Content check — Intermediate Algebra, Chapter 9: Quadratic Equations and Functions

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 9: Quadratic Equations and Functions

434 problems · 105 figures

Be Prepared 9.1 §9.1 source ↗

Simplify:

\sqrt{128} .

If you missed this problem, review Example 8.13.

Example 8.13 in the book ↗

Answer

8 \sqrt{2}

Be Prepared 9.2 §9.1 source ↗

Simplify:

\sqrt{\frac{32}{5}}

.
If you missed this problem, review Example 8.50.

Example 8.50 in the book ↗

Answer

\frac{4 \sqrt{10}}{5}

Be Prepared 9.3 §9.1 source ↗

Factor:

9 x^{2} - 12 x + 4

.
If you missed this problem, review Example 6.23.

Example 6.23 in the book ↗

Answer

{3 x - 2}^{2}

Try It 9.1 §9.1 source ↗

Solve:

x^{2} - 48 = 0 .

Answer

x = 4 \sqrt{3}, x = −4 \sqrt{3}

Try It 9.2 §9.1 source ↗

Solve:

y^{2} - 27 = 0 .

Answer

y = 3 \sqrt{3}, y = −3 \sqrt{3}

Try It 9.3 §9.1 source ↗

Solve:

2 x^{2} = 98 .

Answer

x = 7, x = −7

Try It 9.4 §9.1 source ↗

Solve:

5 m^{2} = 80 .

Answer

m = 4, m = −4

Try It 9.5 §9.1 source ↗

Solve:

c^{2} + 12 = 0 .

Answer

c =

2 \sqrt{3} i,

c =

−2 \sqrt{3} i

Try It 9.6 §9.1 source ↗

Solve:

q^{2} + 24 = 0 .

Answer

c =

2 \sqrt{6} i,

c =

−2 \sqrt{6} i

Try It 9.7 §9.1 source ↗

Solve:

\frac{1}{2} x^{2} + 4 = 24 .

Answer

x =

2 \sqrt{10},

x =

−2 \sqrt{10}

Try It 9.8 §9.1 source ↗

Solve:

\frac{3}{4} y^{2} - 3 = 18 .

Answer

y = 2 \sqrt{7}, y = −2 \sqrt{7}

Try It 9.9 §9.1 source ↗

Solve:

5 r^{2} - 2 = 34 .

Answer

r =

\frac{6 \sqrt{5}}{5},

r =

-

\frac{6 \sqrt{5}}{5}

Try It 9.10 §9.1 source ↗

Solve:

3 t^{2} + 6 = 70 .

Answer

t =

\frac{8 \sqrt{3}}{3},

t =

-

\frac{8 \sqrt{3}}{3}

Try It 9.11 §9.1 source ↗

Solve:

3 {(a - 3)}^{2} = 54 .

Answer

a =

3 +

3 \sqrt{2},

a =

3 -

3 \sqrt{2}

Try It 9.12 §9.1 source ↗

Solve:

2 {(b + 2)}^{2} = 80 .

Answer

b =

−2 +

2 \sqrt{10},

b =

−2 -

2 \sqrt{10}

Try It 9.13 §9.1 source ↗

Solve:

{(x - \frac{1}{2})}^{2} = \frac{5}{4} .

Answer

x = \frac{1}{2} + \frac{\sqrt{5}}{2}

x = \frac{1}{2} - \frac{\sqrt{5}}{2}

Try It 9.14 §9.1 source ↗

Solve:

{(y + \frac{3}{4})}^{2} =

\frac{7}{16} .

Answer

y =

-

\frac{3}{4} +

\frac{\sqrt{7}}{4},

y =

-

\frac{3}{4} -

\frac{\sqrt{7}}{4}

Try It 9.15 §9.1 source ↗

Solve:

5 {(a - 5)}^{2} +

4 =

104 .

Answer

a =

5 +

2 \sqrt{5},

a =

5 -

2 \sqrt{5}

Try It 9.16 §9.1 source ↗

Solve:

3 {(b + 3)}^{2} -

8 =

88 .

Answer

b =

−3 +

4 \sqrt{2},

b =

−3 -

4 \sqrt{2}

Try It 9.17 §9.1 source ↗

Solve:

{(3 r + 4)}^{2} = −8 .

Answer

r =

-

\frac{4}{3} +

\frac{2 \sqrt{2} i}{3},

r =

-

\frac{4}{3} -

\frac{2 \sqrt{2} i}{3}

Try It 9.18 §9.1 source ↗

Solve:

{(2 t - 8)}^{2} =

−10 .

Answer

t =

4 +

\frac{\sqrt{10} i}{2},

t =

4 -

\frac{\sqrt{10} i}{2}

Try It 9.19 §9.1 source ↗

Solve:

9 m^{2} -

12 m +

4 =

25 .

Answer

m = \frac{7}{3}, m = −1

Try It 9.20 §9.1 source ↗

Solve:

16 n^{2} +

40 n +

25 =

4 .

Answer

n = - \frac{3}{4}, n = - \frac{7}{4}

9.1-1 §9.1 source ↗

Solve Quadratic Equations of the Form ax² = k Using the Square Root Property

In the following exercises, solve each equation.

$a^{2} = 49$

Answer

a = \pm 7

9.1-3 §9.1 source ↗

Solve Quadratic Equations of the Form ax² = k Using the Square Root Property

In the following exercises, solve each equation.

$r^{2} - 24 = 0$

Answer

r = \pm 2 \sqrt{6}

9.1-5 §9.1 source ↗

Solve Quadratic Equations of the Form ax² = k Using the Square Root Property

In the following exercises, solve each equation.

$u^{2} - 300 = 0$

Answer

u = \pm 10 \sqrt{3}

9.1-7 §9.1 source ↗

Solve Quadratic Equations of the Form ax² = k Using the Square Root Property

In the following exercises, solve each equation.

$4 m^{2} = 36$

Answer

m = \pm 3

9.1-9 §9.1 source ↗

Solve Quadratic Equations of the Form ax² = k Using the Square Root Property

In the following exercises, solve each equation.

$\frac{4}{3} x^{2} = 48$

Answer

x = \pm 6

9.1-11 §9.1 source ↗

Solve Quadratic Equations of the Form ax² = k Using the Square Root Property

In the following exercises, solve each equation.

$x^{2} + 25 = 0$

Answer

x = \pm 5 i

9.1-13 §9.1 source ↗

Solve Quadratic Equations of the Form ax² = k Using the Square Root Property

In the following exercises, solve each equation.

$x^{2} + 63 = 0$

Answer

x = \pm 3 \sqrt{7} i

9.1-15 §9.1 source ↗

Solve Quadratic Equations of the Form ax² = k Using the Square Root Property

In the following exercises, solve each equation.

$\frac{4}{3} x^{2} + 2 = 110$

Answer

x = \pm 9

9.1-17 §9.1 source ↗

Solve Quadratic Equations of the Form ax² = k Using the Square Root Property

In the following exercises, solve each equation.

$\frac{2}{5} a^{2} + 3 = 11$

Answer

a = \pm 2 \sqrt{5}

9.1-19 §9.1 source ↗

Solve Quadratic Equations of the Form ax² = k Using the Square Root Property

In the following exercises, solve each equation.

$7 p^{2} + 10 = 26$

Answer

p = \pm \frac{4 \sqrt{7}}{7}

9.1-21 §9.1 source ↗

Solve Quadratic Equations of the Form ax² = k Using the Square Root Property

In the following exercises, solve each equation.

$5 y^{2} - 7 = 25$

Answer

y = \pm \frac{4 \sqrt{10}}{5}

9.1-23 §9.1 source ↗

Solve Quadratic Equations of the Form a(x − h)² = k Using the Square Root Property

In the following exercises, solve each equation.

${(u - 6)}^{2} = 64$

Answer

u = 14, u = −2

9.1-25 §9.1 source ↗

Solve Quadratic Equations of the Form a(x − h)² = k Using the Square Root Property

In the following exercises, solve each equation.

${(m - 6)}^{2} = 20$

Answer

m = 6 \pm 2 \sqrt{5}

9.1-27 §9.1 source ↗

Solve Quadratic Equations of the Form a(x − h)² = k Using the Square Root Property

In the following exercises, solve each equation.

${(r - \frac{1}{2})}^{2} = \frac{3}{4}$

Answer

r = \frac{1}{2} \pm \frac{\sqrt{3}}{2}

9.1-29 §9.1 source ↗

Solve Quadratic Equations of the Form a(x − h)² = k Using the Square Root Property

In the following exercises, solve each equation.

${(y + \frac{2}{3})}^{2} = \frac{8}{81}$

Answer

y = - \frac{2}{3} \pm \frac{2 \sqrt{2}}{9}

9.1-31 §9.1 source ↗

Solve Quadratic Equations of the Form a(x − h)² = k Using the Square Root Property

In the following exercises, solve each equation.

${(a - 7)}^{2} + 5 = 55$

Answer

a = 7 \pm 5 \sqrt{2}

9.1-33 §9.1 source ↗

Solve Quadratic Equations of the Form a(x − h)² = k Using the Square Root Property

In the following exercises, solve each equation.

$4 {(x + 3)}^{2} -$ $5 =$ $27$

Answer

x = −3 \pm 2 \sqrt{2}

9.1-35 §9.1 source ↗

Solve Quadratic Equations of the Form a(x − h)² = k Using the Square Root Property

In the following exercises, solve each equation.

${(5 c + 1)}^{2} = −27$

Answer

c = - \frac{1}{5} \pm \frac{3 \sqrt{3}}{5} i

9.1-37 §9.1 source ↗

Solve Quadratic Equations of the Form a(x − h)² = k Using the Square Root Property

In the following exercises, solve each equation.

${(4 x - 3)}^{2} +$ $11 =$ $−17$

Answer

x = \frac{3}{4} \pm \frac{\sqrt{7}}{2} i

9.1-39 §9.1 source ↗

Solve Quadratic Equations of the Form a(x − h)² = k Using the Square Root Property

In the following exercises, solve each equation.

$m^{2} - 4 m + 4 = 8$

Answer

m = 2 \pm 2 \sqrt{2}

9.1-41 §9.1 source ↗

Solve Quadratic Equations of the Form a(x − h)² = k Using the Square Root Property

In the following exercises, solve each equation.

$x^{2} - 6 x + 9 = 12$

Answer

x = 3 \pm 2 \sqrt{3}

9.1-43 §9.1 source ↗

Solve Quadratic Equations of the Form a(x − h)² = k Using the Square Root Property

In the following exercises, solve each equation.

$25 x^{2} -$ $30 x +$ $9 =$ $36$

Answer

x = - \frac{3}{5}, x = \frac{9}{5}

9.1-45 §9.1 source ↗

Solve Quadratic Equations of the Form a(x − h)² = k Using the Square Root Property

In the following exercises, solve each equation.

$36 x^{2} -$ $24 x +$ $4 =$ $81$

Answer

x = - \frac{7}{6}, x = \frac{11}{6}

9.1-47 §9.1 source ↗

In the following exercises, solve using the Square Root Property.

$2 r^{2} = 32$

Answer

r = \pm 4

9.1-49 §9.1 source ↗

In the following exercises, solve using the Square Root Property.

${(a - 4)}^{2} = 28$

Answer

a = 4 \pm 2 \sqrt{7}

9.1-51 §9.1 source ↗

In the following exercises, solve using the Square Root Property.

$9 w^{2} -$ $24 w +$ $16 =$ $1$

Answer

w = 1, w = \frac{5}{3}

9.1-53 §9.1 source ↗

In the following exercises, solve using the Square Root Property.

$a^{2} - 18 = 0$

Answer

a = \pm 3 \sqrt{2}

9.1-55 §9.1 source ↗

In the following exercises, solve using the Square Root Property.

${(p - \frac{1}{3})}^{2} = \frac{7}{9}$

Answer

p = \frac{1}{3} \pm \frac{\sqrt{7}}{3}

9.1-57 §9.1 source ↗

In the following exercises, solve using the Square Root Property.

$m^{2} + 12 = 0$

Answer

m = \pm 2 \sqrt{3} i

9.1-59 §9.1 source ↗

In the following exercises, solve using the Square Root Property.

$u^{2} -$ $14 u +$ $49 =$ $72$

Answer

u = 7 \pm 6 \sqrt{2}

9.1-61 §9.1 source ↗

In the following exercises, solve using the Square Root Property.

${(m - 4)}^{2} + 3 = 15$

Answer

m = 4 \pm 2 \sqrt{3}

9.1-63 §9.1 source ↗

In the following exercises, solve using the Square Root Property.

${(x + 5)}^{2} = 4$

Answer

x = −3, x = −7

9.1-65 §9.1 source ↗

In the following exercises, solve using the Square Root Property.

$6 c^{2} + 4 = 29$

Answer

c = \pm \frac{5 \sqrt{6}}{6}

9.1-67 §9.1 source ↗

In the following exercises, solve using the Square Root Property.

${(x - 6)}^{2} + 7 = 3$

Answer

x = 6 \pm 2 i

9.1-69 §9.1 source ↗

In the following exercises, solve using the Square Root Property.

In your own words, explain the Square Root Property.

Answer

Answers will vary.

Be Prepared 9.4 §9.2 source ↗

Expand:

{(x + 9)}^{2} .

If you missed this problem, review Example 5.32.

Example 5.32 in the book ↗

Answer

x^{2} + 18 x + 81

Be Prepared 9.5 §9.2 source ↗

Factor

y^{2} - 14 y + 49 .

If you missed this problem, review Example 6.9.

Example 6.9 in the book ↗

Answer

(y - 7)^{2}

Be Prepared 9.6 §9.2 source ↗

Factor

5 n^{2} + 40 n + 80 .

If you missed this problem, review Example 6.14.

Example 6.14 in the book ↗

Answer

5 (n + 4)^{2}

Try It 9.21 §9.2 source ↗

Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

ⓐ $a^{2} - 20 a$ ⓑ $m^{2} - 5 m$ ⓒ $p^{2} + \frac{1}{4} p$

Answer

ⓐ

{(a - 10)}^{2}

ⓑ

{(b - \frac{5}{2})}^{2}

ⓒ

{(p + \frac{1}{8})}^{2}

Try It 9.22 §9.2 source ↗

Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

ⓐ $b^{2} - 4 b$ ⓑ $n^{2} + 13 n$ ⓒ $q^{2} - \frac{2}{3} q$

Answer

ⓐ

{(b - 2)}^{2}

ⓑ

{(n + \frac{13}{2})}^{2}

ⓒ

{(q - \frac{1}{3})}^{2}

Try It 9.23 §9.2 source ↗

Solve by completing the square:

x^{2} + 4 x = 5 .

Answer

x = −5, x = 1

Try It 9.24 §9.2 source ↗

Solve by completing the square:

y^{2} - 10 y = −9 .

Answer

y = 1, y = 9

Try It 9.25 §9.2 source ↗

Solve by completing the square:

y^{2} - 10 y = −35 .

Answer

y = 5 \pm \sqrt{10} i

Try It 9.26 §9.2 source ↗

Solve by completing the square:

z^{2} + 8 z = −19 .

Answer

z =

−4 +

\sqrt{3} i,

z =

−4 -

\sqrt{3} i

Try It 9.27 §9.2 source ↗

Solve by completing the square:

x^{2} - 16 x = −16 .

Answer

x =

8 +

4 \sqrt{3},

x =

8 -

4 \sqrt{3}

Try It 9.28 §9.2 source ↗

Solve by completing the square:

y^{2} + 8 y = 11 .

Answer

y =

−4 +

3 \sqrt{3},

y =

−4 -

3 \sqrt{3}

Try It 9.29 §9.2 source ↗

Solve by completing the square:

a^{2} + 4 a + 9 = 30 .

Answer

a = −7, a = 3

Try It 9.30 §9.2 source ↗

Solve by completing the square:

b^{2} + 8 b - 4 = 16 .

Answer

b = −10, b = 2

Try It 9.31 §9.2 source ↗

Solve by completing the square:

p^{2} = 5 p + 9 .

Answer

p =

\frac{5}{2} +

\frac{\sqrt{61}}{2},

p =

\frac{5}{2} -

\frac{\sqrt{61}}{2}

Try It 9.32 §9.2 source ↗

Solve by completing the square:

q^{2} = 7 q - 3 .

Answer

q =

\frac{7}{2} +

\frac{\sqrt{37}}{2},

q =

\frac{7}{2} -

\frac{\sqrt{37}}{2}

Try It 9.33 §9.2 source ↗

Solve by completing the square:

(c - 2) (c + 8) =

11 .

Answer

c = −9, c = 3

Try It 9.34 §9.2 source ↗

Solve by completing the square:

(d - 7) (d + 3) =

56 .

Answer

d = 11, d = −7

Try It 9.35 §9.2 source ↗

Solve by completing the square:

2 m^{2} +

16 m +

14 =

0 .

Answer

m = −7, m = −1

Try It 9.36 §9.2 source ↗

Solve by completing the square:

4 n^{2} -

24 n -

56 =

8 .

Answer

n = −2, n = 8

Try It 9.37 §9.2 source ↗

Solve by completing the square:

3 r^{2} - 2 r = 21 .

Answer

r = - \frac{7}{3}, r = 3

Try It 9.38 §9.2 source ↗

Solve by completing the square:

4 t^{2} + 2 t = 20 .

Answer

t = - \frac{5}{2}, t = 2

Try It 9.39 §9.2 source ↗

Solve by completing the square:

4 x^{2} + 3 x = 2 .

Answer

x =

-

\frac{3}{8} +

\frac{\sqrt{41}}{8},

x =

-

\frac{3}{8} -

\frac{\sqrt{41}}{8}

Try It 9.40 §9.2 source ↗

Solve by completing the square:

3 y^{2} - 10 y = −5 .

Answer

y =

\frac{5}{3} +

\frac{\sqrt{10}}{3},

y =

\frac{5}{3} -

\frac{\sqrt{10}}{3}

9.2-71 §9.2 source ↗

Complete the Square of a Binomial Expression

In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

ⓐ $m^{2} - 24 m$ ⓑ $x^{2} - 11 x$ ⓒ $p^{2} - \frac{1}{3} p$

Answer

ⓐ

{(m - 12)}^{2}

ⓑ

{(x - \frac{11}{2})}^{2}

ⓒ

{(p - \frac{1}{6})}^{2}

9.2-73 §9.2 source ↗

Complete the Square of a Binomial Expression

In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

ⓐ $p^{2} - 22 p$ ⓑ $y^{2} + 5 y$ ⓒ $m^{2} + \frac{2}{5} m$

Answer

ⓐ

{(p - 11)}^{2}

ⓑ

{(y + \frac{5}{2})}^{2}

ⓒ

{(m + \frac{1}{5})}^{2}

9.2-75 §9.2 source ↗

Solve Quadratic Equations of the form x² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

$u^{2} + 2 u = 3$

Answer

u = −3, u = 1

9.2-77 §9.2 source ↗

Solve Quadratic Equations of the form x² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

$x^{2} - 20 x = 21$

Answer

x = −1, x = 21

9.2-79 §9.2 source ↗

Solve Quadratic Equations of the form x² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

$m^{2} + 4 m = −44$

Answer

m = −2 \pm 2 \sqrt{10} i

9.2-81 §9.2 source ↗

Solve Quadratic Equations of the form x² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

$r^{2} + 6 r = −11$

Answer

r = −3 \pm \sqrt{2} i

9.2-83 §9.2 source ↗

Solve Quadratic Equations of the form x² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

$a^{2} - 10 a = −5$

Answer

a = 5 \pm 2 \sqrt{5}

9.2-85 §9.2 source ↗

Solve Quadratic Equations of the form x² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

$x^{2} + 5 x = 2$

Answer

x = - \frac{5}{2} \pm \frac{\sqrt{33}}{2}

9.2-87 §9.2 source ↗

Solve Quadratic Equations of the form x² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

$u^{2} -$ $14 u +$ $12 =$ $−1$

Answer

u = 1, u = 13

9.2-89 §9.2 source ↗

Solve Quadratic Equations of the form x² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

$r^{2} - 4 r - 3 = 9$

Answer

r = −2, r = 6

9.2-91 §9.2 source ↗

Solve Quadratic Equations of the form x² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

$v^{2} = 9 v + 2$

Answer

v = \frac{9}{2} \pm \frac{\sqrt{89}}{2}

9.2-93 §9.2 source ↗

Solve Quadratic Equations of the form x² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

$x^{2} - 5 = 10 x$

Answer

x = 5 \pm \sqrt{30}

9.2-95 §9.2 source ↗

Solve Quadratic Equations of the form x² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

$(x + 6) (x - 2) =$ $9$

Answer

x = −7, x = 3

9.2-97 §9.2 source ↗

Solve Quadratic Equations of the form x² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

$(x + 2) (x + 4) =$ $3$

Answer

x = −5, x = −1

9.2-99 §9.2 source ↗

Solve Quadratic Equations of the form ax² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

$3 m^{2} +$ $30 m -$ $27 =$ $6$

Answer

m = −11, m = 1

9.2-101 §9.2 source ↗

Solve Quadratic Equations of the form ax² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

$2 n^{2} + 4 n = 26$

Answer

n = - 1 \pm \sqrt{14}

9.2-103 §9.2 source ↗

Solve Quadratic Equations of the form ax² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

$2 c^{2} + c = 6$

Answer

c = −2, c = \frac{3}{2}

9.2-105 §9.2 source ↗

Solve Quadratic Equations of the form ax² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

$2 x^{2} + 7 x - 15 = 0$

Answer

x = −5, x = \frac{3}{2}

9.2-107 §9.2 source ↗

Solve Quadratic Equations of the form ax² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

$2 p^{2} + 7 p = 14$

Answer

p = - \frac{7}{4} \pm \frac{\sqrt{161}}{4}

9.2-109 §9.2 source ↗

Solve Quadratic Equations of the form ax² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

$5 x^{2} - 3 x = −10$

Answer

x =

\frac{3}{10} \pm \frac{\sqrt{191}}{10} i

9.2-111 §9.2 source ↗

Solve Quadratic Equations of the form ax² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

Solve the equation $x^{2} + 10 x = −25$

ⓐ by using the Square Root Property

ⓑ by Completing the Square

Answer

Answers will vary.

Be Prepared 9.7 §9.3 source ↗

Evaluate

b^{2} - 4 a b

when

a = 3

and

b = −2 .

If you missed this problem, review Example 1.21.

Example 1.21 in the book ↗

Answer

28

Be Prepared 9.8 §9.3 source ↗

Simplify:

\sqrt{108} .

If you missed this problem, review Example 8.13.

Example 8.13 in the book ↗

Answer

6 \sqrt{3}

Be Prepared 9.9 §9.3 source ↗

Simplify:

\sqrt{50} .

If you missed this problem, review Example 8.76.

Example 8.76 in the book ↗

Answer

5 \sqrt{2}

Try It 9.41 §9.3 source ↗

Solve by using the Quadratic Formula:

3 y^{2} - 5 y + 2 = 0

Answer

y = 1, y = \frac{2}{3}

Try It 9.42 §9.3 source ↗

Solve by using the Quadratic Formula:

4 z^{2} + 2 z - 6 = 0

Answer

z = 1, z = - \frac{3}{2}

Try It 9.43 §9.3 source ↗

Solve by using the Quadratic Formula:

a^{2} - 2 a = 15

Answer

a = −3, a = 5

Try It 9.44 §9.3 source ↗

Solve by using the Quadratic Formula:

b^{2} + 24 = −10 b

Answer

b = −6, b = −4

Try It 9.45 §9.3 source ↗

Solve by using the Quadratic Formula:

3 m^{2} + 12 m + 7 = 0

Answer

m =

\frac{−6 + \sqrt{15}}{3},

m =

\frac{−6 - \sqrt{15}}{3}

Try It 9.46 §9.3 source ↗

Solve by using the Quadratic Formula:

5 n^{2} + 4 n - 4 = 0

Answer

n =

\frac{−2 + 2 \sqrt{6}}{5},

n =

\frac{−2 - 2 \sqrt{6}}{5}

Try It 9.47 §9.3 source ↗

Solve by using the Quadratic Formula:

4 a^{2} - 2 a + 8 = 0

Answer

a =

\frac{1}{4} +

\frac{\sqrt{31}}{4} i,

a =

\frac{1}{4} -

\frac{\sqrt{31}}{4} i

Try It 9.48 §9.3 source ↗

Solve by using the Quadratic Formula:

5 b^{2} + 2 b + 4 = 0

Answer

b =

-

\frac{1}{5} +

\frac{\sqrt{19}}{5} i,

b =

-

\frac{1}{5} -

\frac{\sqrt{19}}{5} i

Try It 9.49 §9.3 source ↗

Solve by using the Quadratic Formula:

x (x + 2) - 5 = 0 .

Answer

x =

−1 +

\sqrt{6},

x =

−1 -

\sqrt{6}

Try It 9.50 §9.3 source ↗

Solve by using the Quadratic Formula:

3 y (y - 2) -

3 =

0 .

Answer

y = 1 + \sqrt{2}, y = 1 - \sqrt{2}

Try It 9.51 §9.3 source ↗

Solve by using the Quadratic Formula:

\frac{1}{4} c^{2} -

\frac{1}{3} c =

\frac{1}{12}

Answer

c =

\frac{2 + \sqrt{7}}{3},

c =

\frac{2 - \sqrt{7}}{3}

Try It 9.52 §9.3 source ↗

Solve by using the Quadratic Formula:

\frac{1}{9} d^{2} -

\frac{1}{2} d =

-

\frac{1}{3}

Answer

d =

\frac{9 + \sqrt{33}}{4},

d =

\frac{9 - \sqrt{33}}{4}

Try It 9.53 §9.3 source ↗

Solve by using the Quadratic Formula:

r^{2} +

10 r +

25 =

0 .

Answer

r = −5

Try It 9.54 §9.3 source ↗

Solve by using the Quadratic Formula:

25 t^{2} -

40 t =

−16 .

Answer

t = \frac{4}{5}

Try It 9.55 §9.3 source ↗

Determine the numberand type of solutions to each quadratic equation.

ⓐ $8 m^{2} - 3 m + 6 = 0$ ⓑ $5 z^{2} + 6 z - 2 = 0$ ⓒ $9 w^{2} +$ $24 w +$ $16 =$ $0 .$

Answer

ⓐ 2 complex solutions; ⓑ 2 real solutions; ⓒ 1 real solution

Try It 9.56 §9.3 source ↗

Determine the number and type of solutions to each quadratic equation.

ⓐ $b^{2} + 7 b - 13 = 0$ ⓑ $5 a^{2} - 6 a + 10 = 0$ ⓒ $4 r^{2} -$ $20 r +$ $25 =$ $0 .$

Answer

ⓐ 2 real solutions; ⓑ 2 complex solutions; ⓒ 1 real solution

Try It 9.57 §9.3 source ↗

Identify the most appropriate method to use to solve each quadratic equation.

ⓐ $x^{2} + 6 x + 8 = 0$ ⓑ ${(n - 3)}^{2} = 16$ ⓒ $5 p^{2} - 6 p = 9 .$

Answer

ⓐ factoring; ⓑ Square Root Property; ⓒ Quadratic Formula

Try It 9.58 §9.3 source ↗

Identify the most appropriate method to use to solve each quadratic equation.

ⓐ $8 a^{2} + 3 a - 9 = 0$ ⓑ $4 b^{2} + 4 b + 1 = 0$ ⓒ $5 c^{2} = 125 .$

Answer

ⓐ Quadratic Forumula;
ⓑ Factoring or Square Root Property ⓒ Square Root Property

9.3-113 §9.3 source ↗

Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

$4 m^{2} + m - 3 = 0$

Answer

m = −1, m = \frac{3}{4}

9.3-115 §9.3 source ↗

Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

$2 p^{2} - 7 p + 3 = 0$

Answer

p = \frac{1}{2}, p = 3

9.3-117 §9.3 source ↗

Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

$p^{2} + 7 p + 12 = 0$

Answer

p = −4, p = −3

9.3-119 §9.3 source ↗

Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

$r^{2} - 8 r = 33$

Answer

r = −3, r = 11

9.3-121 §9.3 source ↗

Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

$3 u^{2} + 7 u - 2 = 0$

Answer

u = \frac{−7 \pm \sqrt{73}}{6}

9.3-123 §9.3 source ↗

Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

$2 a^{2} - 6 a + 3 = 0$

Answer

a = \frac{3 \pm \sqrt{3}}{2}

9.3-125 §9.3 source ↗

Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

$x^{2} + 8 x - 4 = 0$

Answer

x = −4 \pm 2 \sqrt{5}

9.3-127 §9.3 source ↗

Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

$3 y^{2} + 5 y - 2 = 0$

Answer

y = −2, y = \frac{1}{3}

9.3-129 §9.3 source ↗

Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

$2 x^{2} + 3 x + 3 = 0$

Answer

x = - \frac{3}{4} \pm \frac{\sqrt{15}}{4} i

9.3-131 §9.3 source ↗

Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

$8 x^{2} - 6 x + 2 = 0$

Answer

x = \frac{3}{8} \pm \frac{\sqrt{7}}{8} i

9.3-133 §9.3 source ↗

Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

$(v + 1) (v - 5) -$ $4 =$ $0$

Answer

v = 2 \pm \sqrt{13}

9.3-135 §9.3 source ↗

Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

$(y + 4) (y - 7) =$ $18$

Answer

y = \frac{3 \pm \sqrt{193}}{2}

9.3-137 §9.3 source ↗

Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

$\frac{1}{3} m^{2} +$ $\frac{1}{12} m =$ $\frac{1}{4}$

Answer

m = −1, m = \frac{3}{4}

9.3-139 §9.3 source ↗

Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

$\frac{3}{4} b^{2} + \frac{1}{2} b = \frac{3}{8}$

Answer

b = \frac{−2 \pm \sqrt{22}}{6}

9.3-141 §9.3 source ↗

Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

$16 c^{2} +$ $24 c +$ $9 =$ $0$

Answer

c = - \frac{3}{4}

9.3-143 §9.3 source ↗

Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

$25 q^{2} +$ $30 q +$ $9 =$ $0$

Answer

q = - \frac{3}{5}

9.3-145 §9.3 source ↗

Use the Discriminant to Predict the Number of Real Solutions of a Quadratic Equation

In the following exercises, determine the number of real solutions for each quadratic equation.

ⓐ $4 x^{2} - 5 x + 16 = 0$ ⓑ $36 y^{2} +$ $36 y +$ $9 =$ $0$ ⓒ $6 m^{2} + 3 m - 5 = 0$

Answer

ⓐ

no real solutions

ⓑ

1

ⓒ

2

9.3-147 §9.3 source ↗

Use the Discriminant to Predict the Number of Real Solutions of a Quadratic Equation

In the following exercises, determine the number of real solutions for each quadratic equation.

ⓐ $r^{2} + 12 r + 36 = 0$ ⓑ $8 t^{2} - 11 t + 5 = 0$ ⓒ $3 v^{2} - 5 v - 1 = 0$

Answer

ⓐ

1

ⓑ

no real solutions

ⓒ

2

9.3-149 §9.3 source ↗

Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.

ⓐ $x^{2} - 5 x - 24 = 0$
ⓑ ${(y + 5)}^{2} = 12$
ⓒ $14 m^{2} + 3 m = 11$

Answer

ⓐ

factor

ⓑ

square root

ⓒ

Quadratic Formula

9.3-151 §9.3 source ↗

Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.

ⓐ $6 a^{2} + 14 a = 20$
ⓑ ${(x - \frac{1}{4})}^{2} = \frac{5}{16}$
ⓒ $y^{2} - 2 y = 8$

Answer

ⓐ

Quadratic Formula

ⓑ

square root

ⓒ

factor

9.3-153 §9.3 source ↗

Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.

Solve the equation $x^{2} + 10 x = 120$

ⓐ by completing the square

ⓑ using the Quadratic Formula

Answer

Answers will vary.

Be Prepared 9.10 §9.4 source ↗

Factor by substitution:

y^{4} - y^{2} - 20 .

If you missed this problem, review Example 6.21.

Example 6.21 in the book ↗

Answer

y^{2} + 4 y^{2} - 5

Be Prepared 9.11 §9.4 source ↗

Factor by substitution:

{(y - 4)}^{2} +

8 (y - 4) +

15 .

If you missed this problem, review Example 6.22.

Example 6.22 in the book ↗

Answer

(y - 1) (y + 1)

Be Prepared 9.12 §9.4 source ↗

Simplify: ⓐ

x^{\frac{1}{2}} \cdot x^{\frac{1}{4}}

ⓑ

{(x^{\frac{1}{3}})}^{2}

ⓒ

{(x^{−1})}^{2} .

If you missed this problem, review Example 8.33.

Example 8.33 in the book ↗

Answer

ⓐ

x^{\frac{3}{4}}

; ⓑ

x^{\frac{2}{3}}

; ⓒ

x^{- 2}

Try It 9.59 §9.4 source ↗

Solve:

x^{4} - 6 x^{2} + 8 = 0

Answer

x =

\sqrt{2},

x =

− \sqrt{2},

x =

2,

x =

−2

Try It 9.60 §9.4 source ↗

Solve:

x^{4} -

11 x^{2} +

28 =

0

Answer

x =

\sqrt{7},

x =

− \sqrt{7},

x =

2,

x =

−2

Try It 9.61 §9.4 source ↗

Solve:

{(x - 5)}^{2} +

6 (x - 5) +

8 =

0 .

Answer

x = 3, x = 1

Try It 9.62 §9.4 source ↗

Solve:

{(y - 4)}^{2} +

8 (y - 4) +

15 =

0 .

Answer

y = −1, y = 1

Try It 9.63 §9.4 source ↗

Solve:

x - 7 \sqrt{x} + 12 = 0 .

Answer

x = 9, x = 16

Try It 9.64 §9.4 source ↗

Solve:

x - 6 \sqrt{x} + 8 = 0 .

Answer

x = 4, x = 16

Try It 9.65 §9.4 source ↗

Solve:

x^{\frac{2}{3}} -

5 x^{\frac{1}{3}} -

14 =

0 .

Answer

x = −8, x = 343

Try It 9.66 §9.4 source ↗

Solve:

x^{\frac{1}{2}} -

8 x^{\frac{1}{4}} +

15 =

0 .

Answer

x = 81, x = 625

Try It 9.67 §9.4 source ↗

Solve:

8 x^{−2} -

10 x^{−1} +

3 =

0 .

Answer

x = \frac{4}{3} x = 2

Try It 9.68 §9.4 source ↗

Solve:

6 x^{−2} -

23 x^{−1} +

20 =

0 .

Answer

x = \frac{2}{5}, x = \frac{3}{4}

9.4-155 §9.4 source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

$x^{4} - 7 x^{2} + 12 = 0$

Answer

x = \pm \sqrt{3}, x = \pm 2

9.4-157 §9.4 source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

$x^{4} -$ $13 x^{2} -$ $30 =$ $0$

Answer

x = \pm \sqrt{15}, x = \pm \sqrt{2} i

9.4-159 §9.4 source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

$2 x^{4} - 5 x^{2} + 3 = 0$

Answer

x = \pm 1, x = \frac{\pm \sqrt{6}}{2}

9.4-161 §9.4 source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

$2 x^{4} - 7 x^{2} + 3 = 0$

Answer

x = \pm \sqrt{3}, x = \pm \frac{\sqrt{2}}{2}

9.4-163 §9.4 source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

${(x - 3)}^{2} -$ $5 (x - 3) -$ $36 =$ $0$

Answer

x = −1, x = 12

9.4-165 §9.4 source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

${(3 y + 2)}^{2} +$ $(3 y + 2) -$ $6 =$ $0$

Answer

x = - \frac{5}{3}, x = 0

9.4-167 §9.4 source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

${(x^{2} + 1)}^{2} -$ $5 (x^{2} + 1) +$ $4 =$ $0$

Answer

x = 0, x = \pm \sqrt{3}

9.4-169 §9.4 source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

$2 {(x^{2} - 5)}^{2} -$ $5 (x^{2} - 5) +$ $2 =$ $0$

Answer

x = \pm \frac{\sqrt{22}}{2}, x = \pm \sqrt{7}

9.4-171 §9.4 source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

$x - \sqrt{x} - 20 = 0$

Answer

x = 25

9.4-173 §9.4 source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

$x + 6 \sqrt{x} - 16 = 0$

Answer

x = 4

9.4-175 §9.4 source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

$6 x + \sqrt{x} - 2 = 0$

Answer

x = \frac{1}{4}

9.4-177 §9.4 source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

$10 x - 17 \sqrt{x} + 3 = 0$

Answer

x = \frac{1}{25}, x = \frac{9}{4}

9.4-179 §9.4 source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

$x^{\frac{2}{3}} +$ $9 x^{\frac{1}{3}} +$ $8 =$ $0$

Answer

x = −1, x = −512

9.4-181 §9.4 source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

$x^{\frac{2}{3}} + 4 x^{\frac{1}{3}} = 12$

Answer

x = 8, x = −216

9.4-183 §9.4 source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

$6 x^{\frac{2}{3}} - x^{\frac{1}{3}} = 12$

Answer

x =

\frac{27}{8},

x =

-

\frac{64}{27}

9.4-185 §9.4 source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

$8 x^{\frac{2}{3}} -$ $43 x^{\frac{1}{3}} +$ $15 =$ $0$

Answer

x =

\frac{27}{512},

x =

125

9.4-187 §9.4 source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

$x - 8 x^{\frac{1}{2}} + 7 = 0$

Answer

x = 1, x = 49

9.4-189 §9.4 source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

$6 x^{−2} +$ $13 x^{−1} +$ $5 =$ $0$

Answer

x = −2, x = - \frac{3}{5}

9.4-191 §9.4 source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

$8 x^{−2} -$ $2 x^{−1} -$ $3 =$ $0$

Answer

x = −2, x = \frac{4}{3}

9.4-193 §9.4 source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

Explain how to recognize an equation in quadratic form.

Answer

Answers will vary.

Be Prepared 9.13 §9.5 source ↗

The sum of two consecutive odd numbers is −100. Find the numbers.
If you missed this problem, review Example 2.18.

Example 2.18 in the book ↗

Answer

- 51, - 49

Be Prepared 9.14 §9.5 source ↗

Solve:

\frac{2}{x + 1} +

\frac{1}{x - 1} =

\frac{1}{x^{2} - 1} .

If you missed this problem, review Example 7.35.

Example 7.35 in the book ↗

Answer

x = \frac{2}{3}

Be Prepared 9.15 §9.5 source ↗

Find the length of the hypotenuse of a right triangle with legs 5 inches and 12 inches.
If you missed this problem, review Example 2.34.

Example 2.34 in the book ↗

Answer

13 inches

Try It 9.69 §9.5 source ↗

The product of two consecutive odd integers is 99. Find the integers.

Answer

The two consecutive odd integers whose product is 99 are 9, 11, and −9, −11

Try It 9.70 §9.5 source ↗

The product of two consecutive even integers is 168. Find the integers.

Answer

The two consecutive even integers whose product is 128 are 12, 14 and −12, −14.

Try It 9.71 §9.5 source ↗

Find the base and height of a triangle whose base is four inches more than six times its height and has an area of 456 square inches.

Answer

The height of the triangle is 12 inches and the base is 76 inches.

Try It 9.72 §9.5 source ↗

If a triangle that has an area of 110 square feet has a base that is two feet less than twice the height, what is the length of its base and height?

Answer

The height of the triangle is 11 feet and the base is 20 feet.

Try It 9.73 §9.5 source ↗

The length of a 200 square foot rectangular vegetable garden is four feet less than twice the width. Find the length and width of the garden, to the nearest tenth of a foot.

Answer

The length of the garden is approximately 18 feet and the width 11 feet.

Try It 9.74 §9.5 source ↗

A rectangular tablecloth has an area of 80 square feet. The width is 5 feet shorter than the length.What are the length and width of the tablecloth to the nearest tenth of a foot.?

Answer

The length of the tablecloth is approximatel 11.8 feet and the width 6.8 feet.

Try It 9.75 §9.5 source ↗

The sun casts a shadow from a flag pole. The height of the flag pole is three times the length of its shadow. The distance between the end of the shadow and the top of the flag pole is 20 feet. Find the length of the shadow and the length of the flag pole. Round to the nearest tenth.

Answer

The length of the flag pole’s shadow is approximately 6.3 feet and the height of the flag pole is 18.9 feet.

Try It 9.76 §9.5 source ↗

The distance between opposite corners of a rectangular field is four more than the width of the field. The length of the field is twice its width. Find the distance between the opposite corners. Round to the nearest tenth.

Answer

The distance between the opposite corners is approximately 7.2 feet.

Try It 9.77 §9.5 source ↗

An arrow is shot from the ground into the air at an initial speed of 108 ft/s. Use the formula h = −16t² + v₀t to determine when the arrow will be 180 feet from the ground. Round the nearest tenth.

Answer

The arrow will reach 180 feet on its way up after 3 seconds and again on its way down after approximately 3.8 seconds.

Try It 9.78 §9.5 source ↗

A man throws a ball into the air with a velocity of 96 ft/s. Use the formula h = −16t² + v₀t to determine when the height of the ball will be 48 feet. Round to the nearest tenth.

Answer

The ball will reach 48 feet on its way up after approximately .6 second and again on its way down after approximately 5.4 seconds.

Try It 9.79 §9.5 source ↗

MaryAnne just returned from a visit with her grandchildren back east . The trip was 2400 miles from her home and her total time in the airplane for the round trip was 10 hours. If the plane was flying at a rate of 500 miles per hour, what was the speed of the jet stream?

Answer

The speed of the jet stream was 100 mph.

Try It 9.80 §9.5 source ↗

Gerry just returned from a cross country trip. The trip was 3000 miles from his home and his total time in the airplane for the round trip was 11 hours. If the plane was flying at a rate of 550 miles per hour, what was the speed of the jet stream?

Answer

The speed of the jet stream was 50 mph.

Try It 9.81 §9.5 source ↗

The weekly news magazine has a big story naming the Person of the Year and the editor wants the magazine to be printed as soon as possible. She has asked the printer to run an extra printing press to get the printing done more quickly. Press #1 takes 6 hours more than Press #2 to do the job and when both presses are running they can print the job in 4 hours. How long does it take for each press to print the job alone?

Answer

Press #1 would take 12 hours, and Press #2 would take 6 hours to do the job alone.

Try It 9.82 §9.5 source ↗

Erlinda is having a party and wants to fill her hot tub. If she only uses the red hose it takes 3 hours more than if she only uses the green hose. If she uses both hoses together, the hot tub fills in 2 hours. How long does it take for each hose to fill the hot tub?

Answer

The red hose take 6 hours and the green hose take 3 hours alone.

9.5-195 §9.5 source ↗

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve using any method.

The product of two consecutive odd numbers is 255. Find the numbers.

Answer

Two consecutive odd numbers whose product is 255 are 15 and 17, and −15 and −17.

9.5-197 §9.5 source ↗

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve using any method.

The product of two consecutive even numbers is 624. Find the numbers.

Answer

The first and second consecutive odd numbers are 24 and 26, and −26 and −24.

9.5-199 §9.5 source ↗

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve using any method.

The product of two consecutive odd numbers is 483. Find the numbers.

Answer

Two consecutive odd numbers whose product is 483 are 21 and 23, and −21 and −23.

9.5-201 §9.5 source ↗

In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.

A triangle with area 45 square inches has a height that is two less than four times the base Find the base and height of the triangle.

Answer

The width of the triangle is 5 inches and the height is 18 inches.

9.5-203 §9.5 source ↗

In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.

The area of a triangular flower bed in the park has an area of 120 square feet. The base is 4 feet longer that twice the height. What are the base and height of the triangle?

Answer

The base is 24 feet and the height of the triangle is 10 feet.

9.5-205 §9.5 source ↗

In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.

The length of a rectangular driveway is five feet more than three times the width. The area is 50 square feet. Find the length and width of the driveway.

Answer

The length of the driveway is 15.0 feet and the width is 3.3 feet.

9.5-207 §9.5 source ↗

In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.

A rectangular table for the dining room has a surface area of 24 square feet. The length is two more feet than twice the width of the table. Find the length and width of the table.

Answer

The length of table is 8 feet and the width is 3 feet.

9.5-209 §9.5 source ↗

In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.

The hypotenuse of a right triangle is twice the length of one of its legs. The length of the other leg is three feet. Find the lengths of the three sides of the triangle.

Answer

The lengths of the three sides of the triangle are 1.7, 3, and 3.5 ft.

9.5-211 §9.5 source ↗

In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.

A rectangular garden will be divided into two plots by fencing it on the diagonal. The diagonal distance from one corner of the garden to the opposite corner is five yards longer than the width of the garden. The length of the garden is three times the width. Find the length of the diagonal of the garden.

Image shows a rectangular segment of grass with fence around 4 sides and across the diagonal. The vertical side of the rectangle is labeled w and the horizontal side is labeled 3 w. The diagonal fence is labeled w plus 5.

Answer

The length of the diagonal fencing is 7.3 yards.

9.5-213 §9.5 source ↗

In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.

Gerry plans to place a 25-foot ladder against the side of his house to clean his gutters. The bottom of the ladder will be 5 feet from the house.How far up the side of the house will the ladder reach?

Answer

The ladder will reach 24.5 feet on the side of the house.

9.5-215 §9.5 source ↗

In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.

A firework rocket is shot upward at a rate of 640 ft/sec. Use the projectile formula h = −16t² + v₀t to determine when the height of the firework rocket will be 1200 feet.

Answer

The rocket will reach 1200 feet on its way up at 1.97 seconds and on its way down at 38.03 seconds.

9.5-217 §9.5 source ↗

In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.

A bullet is fired straight up from a BB gun with initial velocity 1120 feet per second at an initial height of 8 feet. Use the formula h = −16t² + v₀t + 8 to determine how many seconds it will take for the bullet to hit the ground. (That is, when will h = 0?)

Answer

The bullet will take 70 seconds to hit the ground.

9.5-219 §9.5 source ↗

In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.

The businessman took a small airplane for a quick flight up the coast for a lunch meeting and then returned home. The plane flew a total of 4 hours and each way the trip was 200 miles. What was the speed of the wind that affected the plane which was flying at a speed of 120 mph?

Answer

The speed of the wind was 49 mph.

9.5-221 §9.5 source ↗

In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.

Roy kayaked up the river and then back in a total time of 6 hours. The trip was 4 miles each way and the current was difficult. If Roy kayaked at a speed of 5 mph, what was the speed of the current?

Answer

The speed of the current was 4.3 mph.

9.5-223 §9.5 source ↗

In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.

Two painters can paint a room in 2 hours if they work together. The less experienced painter takes 3 hours more than the more experienced painter to finish the job. How long does it take for each painter to paint the room individually?

Answer

The less experienced painter takes 6 hours and the experienced painter takes 3 hours to do the job alone.

9.5-225 §9.5 source ↗

In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.

It takes two hours for two machines to manufacture 10,000 parts. If Machine #1 can do the job alone in one hour less than Machine #2 can do the job, how long does it take for each machine to manufacture 10,000 parts alone?

Answer

Machine #1 takes 3.6 hours and Machine #2 takes 4.6 hours to do the job alone.

9.5-227 §9.5 source ↗

In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.

Make up a problem involving the product of two consecutive odd integers.

ⓐ Start by choosing two consecutive odd integers. What are your integers?

ⓑ What is the product of your integers?

ⓓ Did you get the numbers you started with?

Answer

Answers will vary.

Be Prepared 9.16 §9.6 source ↗

Graph the function

f (x) = x^{2}

by plotting points.
If you missed this problem, review Example 3.54.

Example 3.54 in the book ↗

Answer

An upward-opening parabola graphed on a Cartesian coordinate system, with its vertex at (0,0) and key points highlighted.

Be Prepared 9.17 §9.6 source ↗

Solve:

2 x^{2} + 3 x - 2 = 0 .

If you missed this problem, review Example 6.45.

Example 6.45 in the book ↗

Answer

x = \frac{1}{2}, x = - 2

Be Prepared 9.18 §9.6 source ↗

Evaluate

- \frac{b}{2 a}

when a = 3 and b = −6.
If you missed this problem, review Example 1.21.

Example 1.21 in the book ↗

Answer

1

Try It 9.83 §9.6 source ↗

Graph

f (x) = − x^{2} .

Answer

Try It 9.84 §9.6 source ↗

Graph

f (x) = x^{2} + 1 .

Answer

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (0, −1).

Try It 9.85 §9.6 source ↗

Determine whether the graph of each function is a parabola that opens upward or downward:

ⓐ $f (x) =$ $2 x^{2} +$ $5 x -$ $2$ ⓑ $f (x) =$ $−3 x^{2} -$ $4 x +$ $7 .$

Answer

ⓐ up; ⓑ down

Try It 9.86 §9.6 source ↗

Determine whether the graph of each function is a parabola that opens upward or downward:

ⓐ $f (x) =$ $−2 x^{2} -$ $2 x -$ $3$ ⓑ $f (x) =$ $5 x^{2} -$ $2 x -$ $1 .$

Answer

ⓐ down; ⓑ up

Try It 9.87 §9.6 source ↗

For the graph of $f (x) =$ $2 x^{2} -$ $8 x +$ $1$ find:

ⓐ the axis of symmetry ⓑ the vertex.

Answer

ⓐ

x = 2;

ⓑ

(2, −7)

Try It 9.88 §9.6 source ↗

For the graph of $f (x) =$ $2 x^{2} -$ $4 x -$ $3$ find:

ⓐ the axis of symmetry ⓑ the vertex.

Answer

ⓐ

x = 1;

ⓑ

(1, −5)

Try It 9.89 §9.6 source ↗

Find the intercepts of the parabola whose function is

f (x) =

x^{2} +

2 x -

8 .

Answer

y-intercept:

(0, −8)

x-intercepts

(−4, 0),

(2, 0)

Try It 9.90 §9.6 source ↗

Find the intercepts of the parabola whose function is

f (x) =

x^{2} -

4 x -

12 .

Answer

y-intercept:

(0, −12)

x-intercepts

(−2, 0),

(6, 0)

Try It 9.91 §9.6 source ↗

Find the intercepts of the parabola whose function is

f (x) =

3 x^{2} +

4 x +

4 .

Answer

y-intercept:

(0, 4)

no x-intercept

Try It 9.92 §9.6 source ↗

Find the intercepts of the parabola whose function is

f (x) =

x^{2} -

4 x -

5 .

Answer

y-intercept:

(0, −5)

x-intercepts

(−1, 0),

(5, 0)

Try It 9.93 §9.6 source ↗

Graph f (x) = x² + 2x − 8 by using its properties.

Answer

Try It 9.94 §9.6 source ↗

Graph f (x) = x² − 8x + 12 by using its properties.

Answer

Try It 9.95 §9.6 source ↗

Graph f (x) = −3x² + 12x − 12 by using its properties.

Answer

This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 15 to 10. The parabola has a vertex at (2, 0). The y-intercept (0, negative 12) is plotted as well as the axis of symmetry, x equals 2.

Try It 9.96 §9.6 source ↗

Graph f (x) = 4x² + 24x + 36 by using its properties.

Answer

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 30 to 20. The y-axis of the plane runs from negative 10 to 40. The parabola has a vertex at (negative 3, 0). The y-intercept (0, 36) is plotted as well as the axis of symmetry, x equals negative 3.

Try It 9.97 §9.6 source ↗

Graph f (x) = x² − 2x + 3 by using its properties.

Answer

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 4. The y-axis of the plane runs from negative 1 to 5. The parabola has a vertex at (1, 2). The y-intercept (0, 3) is plotted as is the line of symmetry, x equals 1.

Try It 9.98 §9.6 source ↗

Graph f (x) = −3x² − 6x − 4 by using its properties.

Answer

This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 2. The y-axis of the plane runs from negative 5 to 1. The parabola has a vertex at (negative 1, negative 2). The y-intercept (0, negative 4) is plotted as is the line of symmetry, x equals negative 1.

Try It 9.99 §9.6 source ↗

Graph f (x) = 5x² + 10x + 3 by using its properties.

Answer

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis of the plane runs from negative 4 to 4. The axis of symmetry, x equals negative 1, is graphed as a dashed line. The parabola has a vertex at (negative 1, negative 2). The y-intercept of the parabola is the point (0, 3). The x-intercepts of the parabola are approximately (negative 1.6, 0) and (negative 0.4, 0).

Try It 9.100 §9.6 source ↗

Graph f (x) = −3x² − 6x + 5 by using its properties.

Answer

Try It 9.101 §9.6 source ↗

Find the maximum or minimum value of the quadratic function

f (x) =

x^{2} -

8 x +

12 .

Answer

The minimum value of the quadratic function is −4 and it occurs when x = 4.

Try It 9.102 §9.6 source ↗

Find the maximum or minimum value of the quadratic function

f (x) =

−4 x^{2} +

16 x -

11 .

Answer

The maximum value of the quadratic function is 5 and it occurs when x = 2.

Try It 9.103 §9.6 source ↗

Solve, rounding answers to the nearest tenth.

The quadratic function h(t) = −16t² + 128t + 32 is used to find the height of a stone thrown upward from a height of 32 feet at a rate of 128 ft/sec. How long will it take for the stone to reach its maximum height? What is the maximum height?

Answer

It will take 4 seconds for the stone to reach its maximum height of 288 feet.

Try It 9.104 §9.6 source ↗

A path of a toy rocket thrown upward from the ground at a rate of 208 ft/sec is modeled by the quadratic function of h(t) = −16t² + 208t. When will the rocket reach its maximum height? What will be the maximum height?

Answer

It will 6.5 seconds for the rocket to reach its maximum height of 676 feet.

9.6-229 §9.6 source ↗

Recognize the Graph of a Quadratic Function

In the following exercises, graph the functions by plotting points.

$f (x) = x^{2} + 3$

Answer

9.6-231 §9.6 source ↗

Recognize the Graph of a Quadratic Function

In the following exercises, graph the functions by plotting points.

$y = − x^{2} + 1$

Answer

9.6-233 §9.6 source ↗

For each of the following exercises, determine if the parabola opens up or down.

ⓐ $f (x) =$ $−2 x^{2} -$ $6 x -$ $7$ ⓑ $f (x) =$ $6 x^{2} +$ $2 x +$ $3$

Answer

ⓐ down ⓑ up

9.6-235 §9.6 source ↗

For each of the following exercises, determine if the parabola opens up or down.

ⓐ $f (x) =$ $−3 x^{2} +$ $5 x -$ $1$ ⓑ $f (x) =$ $2 x^{2} -$ $4 x +$ $5$

Answer

ⓐ down ⓑ up

9.6-237 §9.6 source ↗

Find the Axis of Symmetry and Vertex of a Parabola

In the following functions, find ⓐ the equation of the axis of symmetry and ⓑ the vertex of its graph.

$f (x) =$ $x^{2} +$ $8 x -$ $1$

Answer

ⓐ

x = −4

; ⓑ

(−4, −17)

9.6-239 §9.6 source ↗

Find the Axis of Symmetry and Vertex of a Parabola

In the following functions, find ⓐ the equation of the axis of symmetry and ⓑ the vertex of its graph.

$f (x) =$ $− x^{2} +$ $2 x +$ $5$

Answer

ⓐ

x = 1

; ⓑ

(1, 6)

9.6-241 §9.6 source ↗

Find the Intercepts of a Parabola

In the following exercises, find the intercepts of the parabola whose function is given.

$f (x) =$ $x^{2} +$ $7 x +$ $6$

Answer

y-intercept:

(0, 6);

x-intercept

(−1, 0),

(−6, 0)

9.6-243 §9.6 source ↗

Find the Intercepts of a Parabola

In the following exercises, find the intercepts of the parabola whose function is given.

$f (x) =$ $x^{2} +$ $8 x +$ $12$

Answer

y-intercept:

(0, 12);

x-intercept

(−2, 0),

(−6, 0)

9.6-245 §9.6 source ↗

Find the Intercepts of a Parabola

In the following exercises, find the intercepts of the parabola whose function is given.

$f (x) =$ $− x^{2} +$ $8 x -$ $19$

Answer

y-intercept:

(0, −19);

x-intercept: none

9.6-247 §9.6 source ↗

Find the Intercepts of a Parabola

In the following exercises, find the intercepts of the parabola whose function is given.

$f (x) =$ $x^{2} +$ $6 x +$ $13$

Answer

y-intercept:

(0, 13);

x-intercept: none

9.6-249 §9.6 source ↗

Find the Intercepts of a Parabola

In the following exercises, find the intercepts of the parabola whose function is given.

$f (x) =$ $4 x^{2} -$ $20 x +$ $25$

Answer

y-intercept:

(0, 25);

x-intercept

(\frac{5}{2}, 0)

9.6-251 §9.6 source ↗

Find the Intercepts of a Parabola

In the following exercises, find the intercepts of the parabola whose function is given.

$f (x) =$ $− x^{2} -$ $6 x -$ $9$

Answer

y-intercept:

(0, −9);

x-intercept

(−3, 0)

9.6-253 §9.6 source ↗

Graph Quadratic Functions Using Properties

In the following exercises, graph the function by using its properties.

$f (x) =$ $x^{2} +$ $6 x +$ $5$

Answer

9.6-255 §9.6 source ↗

Graph Quadratic Functions Using Properties

In the following exercises, graph the function by using its properties.

$f (x) =$ $x^{2} +$ $4 x +$ $3$

Answer

9.6-257 §9.6 source ↗

Graph Quadratic Functions Using Properties

In the following exercises, graph the function by using its properties.

$f (x) =$ $9 x^{2} +$ $12 x +$ $4$

Answer

9.6-259 §9.6 source ↗

Graph Quadratic Functions Using Properties

In the following exercises, graph the function by using its properties.

$f (x) =$ $− x^{2} +$ $2 x -$ $7$

Answer

9.6-261 §9.6 source ↗

Graph Quadratic Functions Using Properties

In the following exercises, graph the function by using its properties.

$f (x) =$ $2 x^{2} -$ $4 x +$ $1$

Answer

9.6-263 §9.6 source ↗

Graph Quadratic Functions Using Properties

In the following exercises, graph the function by using its properties.

$f (x) =$ $2 x^{2} -$ $4 x +$ $2$

Answer

9.6-265 §9.6 source ↗

Graph Quadratic Functions Using Properties

In the following exercises, graph the function by using its properties.

$f (x) =$ $− x^{2} -$ $4 x +$ $2$

Answer

9.6-267 §9.6 source ↗

Graph Quadratic Functions Using Properties

In the following exercises, graph the function by using its properties.

$f (x) =$ $5 x^{2} -$ $10 x +$ $8$

Answer

9.6-269 §9.6 source ↗

Graph Quadratic Functions Using Properties

In the following exercises, graph the function by using its properties.

$f (x) =$ $3 x^{2} +$ $18 x +$ $20$

Answer

9.6-271 §9.6 source ↗

Solve Maximum and Minimum Applications

In the following exercises, find the maximum or minimum value of each function.

$f (x) =$ $2 x^{2} +$ $x -$ $1$

Answer

The minimum value is

- \frac{9}{8}

when

x = - \frac{1}{4} .

9.6-273 §9.6 source ↗

Solve Maximum and Minimum Applications

In the following exercises, find the maximum or minimum value of each function.

$y = x^{2} - 6 x + 15$

Answer

The minimum value is 6 when x = 3.

9.6-275 §9.6 source ↗

Solve Maximum and Minimum Applications

In the following exercises, find the maximum or minimum value of each function.

$y = −9 x^{2} + 16$

Answer

The maximum value is 16 when x = 0.

9.6-277 §9.6 source ↗

In the following exercises, solve. Round answers to the nearest tenth.

An arrow is shot vertically upward from a platform 45 feet high at a rate of 168 ft/sec. Use the quadratic function h(t) = −16t² + 168t + 45 find how long it will take the arrow to reach its maximum height, and then find the maximum height.

Answer

In 5.3 sec the arrow will reach maximum height of 486 ft.

9.6-279 §9.6 source ↗

In the following exercises, solve. Round answers to the nearest tenth.

A ball is thrown vertically upward from the ground with an initial velocity of 109 ft/sec. Use the quadratic function h(t) = −16t² + 109t + 0 to find how long it will take for the ball to reach its maximum height, and then find the maximum height.

Answer

In 3.4 seconds the ball will reach its maximum height of 185.6 feet.

9.6-281 §9.6 source ↗

In the following exercises, solve. Round answers to the nearest tenth.

A computer store owner estimates that by charging x dollars each for a certain computer, he can sell 40 − x computers each week. The quadratic function R(x) = −x² +40x is used to find the revenue, R, received when the selling price of a computer is x, Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue.

Answer

A selling price of $20 per computer will give the maximum revenue of $400.

9.6-283 §9.6 source ↗

In the following exercises, solve. Round answers to the nearest tenth.

A retailer who sells fashion boots estimates that by selling them for x dollars each, he will be able to sell 70 − x boots a week. Use the quadratic function R(x) = −x² +70x to find the revenue received when the average selling price of a pair of fashion boots is x. Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue per day.

Answer

A selling price of $35 per pair of boots will give a maximum revenue of $1,225.

9.6-285 §9.6 source ↗

In the following exercises, solve. Round answers to the nearest tenth.

A rancher is going to fence three sides of a corral next to a river. He needs to maximize the corral area using 240 feet of fencing. The quadratic equation $A (x) =$ $x 120 - \frac{x}{2}$ gives the area of the corral, A, for the length, x, of the corral along the river. Find the length of the corral along the river that will give the maximum area, and then find the maximum area of the corral.

Answer

The length of one side along the river is 120 feet and the maximum are is 7,200 square feet.

9.6-287 §9.6 source ↗

In the following exercises, solve. Round answers to the nearest tenth.

A land owner is planning to build a fenced in rectangular patio behind his garage, using his garage as one of the “walls.” He wants to maximize the area using 80 feet of fencing. The quadratic function A(x) = x(80 − 2x) gives the area of the patio, where x is the width of one side. Find the maximum area of the patio.

Answer

The maximum area of the patio is 800 feet.

9.6-289 §9.6 source ↗

In the following exercises, solve. Round answers to the nearest tenth.

How do the graphs of the functions $f (x) = x^{2}$ and $f (x) = x^{2} - 1$ differ? We graphed them at the start of this section. What is the difference between their graphs? How are their graphs the same?

Answer

Answers will vary.

9.6-291 §9.6 source ↗

In the following exercises, solve. Round answers to the nearest tenth.

Explain how to find the intercepts of a parabola.

Answer

Answers will vary.

Be Prepared 9.19 §9.7 source ↗

Graph the function

f (x) = x^{2}

by plotting points.
If you missed this problem, review Example 3.54.

Example 3.54 in the book ↗

Answer

A parabola opening upwards, with its vertex at the origin (0,0) and symmetrical about the y-axis, is displayed on a Cartesian coordinate system with grid lines from -8 to 8 on both axes.

Be Prepared 9.20 §9.7 source ↗

Factor completely:

y^{2} - 14 y + 49 .

If you missed this problem, review Example 6.24.

Example 6.24 in the book ↗

Answer

{y^{- 7}}^{2}

Be Prepared 9.21 §9.7 source ↗

Factor completely:

2 x^{2} - 16 x + 32 .

If you missed this problem, review Example 6.26.

Example 6.26 in the book ↗

Answer

2 {(x^{- 4})}^{2}

Try It 9.105 §9.7 source ↗

ⓐ Graph

f (x) =

x^{2},

g (x) =

x^{2} +

1,

and

h (x) = x^{2} - 1

on the same rectangular coordinate system.
ⓑ Describe what effect adding a constant to the function has on the basic parabola.

Answer

ⓐ

This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle graph is of f of x equals x squared has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 1 unit, and the bottom has been moved down 1 unit.

ⓑ The graph of $g (x) = x^{2} + 1$ is the same as the graph of $f (x) = x^{2}$ but shifted up 1 unit. The graph of $h (x) = x^{2} - 1$ is the same as the graph of $f (x) = x^{2}$ but shifted down 1 unit.

Try It 9.106 §9.7 source ↗

ⓐ Graph

f (x) =

x^{2},

g (x) =

x^{2} +

6,

and

h (x) = x^{2} - 6

on the same rectangular coordinate system.
ⓑ Describe what effect adding a constant to the function has on the basic parabola.

Answer

ⓐ

This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 6 units, and the bottom has been moved down 6 units.

ⓑ The graph of $h (x) = x^{2} + 6$ is the same as the graph of $f (x) = x^{2}$ but shifted up 6 units. The graph of $h (x) = x^{2} - 6$ is the same as the graph of $f (x) = x^{2}$ but shifted down 6 units.

Try It 9.107 §9.7 source ↗

Graph

f (x) = x^{2} - 5

using a vertical shift.

Answer

Try It 9.108 §9.7 source ↗

Graph

f (x) = x^{2} + 7

using a vertical shift.

Answer

This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The bottom curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 7 units.

Try It 9.109 §9.7 source ↗

ⓐ Graph

f (x) =

x^{2},

g (x) =

{(x + 2)}^{2},

and

h (x) = {(x - 2)}^{2}

on the same rectangular coordinate system.
ⓑ Describe what effect adding a constant to the function has on the basic parabola.

Answer

ⓐ

ⓑ The graph of $g (x) = {(x + 2)}^{2}$ is the same as the graph of $f (x) = x^{2}$ but shifted left 2 units. The graph of $h (x) = {(x - 2)}^{2}$ is the same as the graph of $f (x) = x^{2}$ but shift right 2 units.

Try It 9.110 §9.7 source ↗

ⓐ Graph

f (x) =

x^{2},

g (x) =

x^{2} +

5,

and

h (x) = x^{2} - 5

on the same rectangular coordinate system.
ⓑ Describe what effect adding a constant to the function has on the basic parabola.

Answer

ⓐ

ⓑ The graph of $g (x) = {(x + 5)}^{2}$ is the same as the graph of $f (x) = x^{2}$ but shifted left 5 units. The graph of $h (x) = {(x - 5)}^{2}$ is the same as the graph of $f (x) = x^{2}$ but shifted right 5 units.

Try It 9.111 §9.7 source ↗

Graph

f (x) = {(x - 4)}^{2}

using a horizontal shift.

Answer

Try It 9.112 §9.7 source ↗

Graph

f (x) = {(x + 6)}^{2}

using a horizontal shift.

Answer

Try It 9.113 §9.7 source ↗

Graph

f (x) =

{(x + 2)}^{2} -

3

using transformations.

Answer

Try It 9.114 §9.7 source ↗

Graph

f (x) =

{(x - 3)}^{2} +

1

using transformations.

Answer

Try It 9.115 §9.7 source ↗

Graph

f (x) = −3 x^{2} .

Answer

Try It 9.116 §9.7 source ↗

Graph

f (x) = 2 x^{2} .

Answer

This figure shows 2 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The slimmer curve of f of x equals 2 times x square has a vertex at (0,0) and other points of (negative 1, one-half) and (1, one-half).

Try It 9.117 §9.7 source ↗

Rewrite

f (x) =

−4 x^{2} -

8 x +

1

in the

f (x) =

a {(x - h)}^{2} +

k

form by completing the square.

Answer

f (x) =

−4 {(x + 1)}^{2} +

5

Try It 9.118 §9.7 source ↗

Rewrite

f (x) =

2 x^{2} -

8 x +

3

in the

f (x) =

a {(x - h)}^{2} +

k

form by completing the square.

Answer

f (x) =

2 {(x - 2)}^{2} -

5

Try It 9.119 §9.7 source ↗

Graph

f (x) =

x^{2} +

2 x -

3

by using transformations.

Answer

Try It 9.120 §9.7 source ↗

Graph

f (x) =

x^{2} -

8 x +

12

by using transformations.

Answer

Try It 9.121 §9.7 source ↗

Graph

f (x) =

−3 x^{2} +

12 x -

4

by using transformations.

Answer

This figure shows a downward-opening parabola on the x y-coordinate plane with a vertex of (2,8) and other points of (1,5) and (3,5).

Try It 9.122 §9.7 source ↗

Graph

f (x) =

−2 x^{2} +

12 x -

9

by using transformations.

Answer

This figure shows a downward-opening parabola on the x y-coordinate plane with a vertex of (3, 9) and other points of (1, 1) and (5, 1).

Try It 9.123 §9.7 source ↗

ⓐ Rewrite

f (x) =

3 x^{2} -

6 x +

5

f (x) =

a {(x - h)}^{2} +

k

form and ⓑ graph the function using properties.

Answer

ⓐ $f (x) =$ $3 {(x - 1)}^{2} +$ $2$
ⓑ

The graph shown is an upward facing parabola with vertex (1, 2) and y-intercept (0, 5). The axis of symmetry is shown, x equals 1.

Try It 9.124 §9.7 source ↗

ⓐ Rewrite

f (x) =

−2 x^{2} +

8 x -

7

f (x) =

a {(x - h)}^{2} +

k

form and ⓑ graph the function using properties.

Answer

ⓐ $f (x) =$ $−2 {(x - 2)}^{2} +$ $1$
ⓑ

The graph shown is a downward facing parabola with vertex (2, 1) and x-intercepts (1, 0) and (3, 0). The axis of symmetry is shown, x equals 2.

Try It 9.125 §9.7 source ↗

Write the quadratic function in $f (x) =$ $a {(x - h)}^{2} +$ $k$ form whose graph is shown.

The graph shown is an upward facing parabola with vertex (3, negative 4) and y-intercept (0, 5).

Answer

f (x) =

{(x - 3)}^{2} -

4

Try It 9.126 §9.7 source ↗

Determine the quadratic function whose graph is shown.

The graph shown is an upward facing parabola with vertex (negative 3, negative 1) and y-intercept (0, 8).

Answer

f (x) =

{(x + 3)}^{2} -

1

9.7-293 §9.7 source ↗

Graph Quadratic Functions of the form $f (x) = x^{2} + k$

In the following exercises, ⓐ graph the quadratic functions on the same rectangular coordinate system and ⓑ describe what effect adding a constant, k, to the function has on the basic parabola.

$f (x) =$ $x^{2},$ $g (x) =$ $x^{2} +$ $4,$ and $h (x) = x^{2} - 4 .$

Answer

ⓐ

ⓑ The graph of $g (x) = x^{2} + 4$ is the same as the graph of $f (x) = x^{2}$ but shifted up 4 units. The graph of $h (x) = x^{2} - 4$ is the same as the graph of $f (x) = x^{2}$ but shift down 4 units.

9.7-295 §9.7 source ↗

In the following exercises, graph each function using a vertical shift.

$f (x) = x^{2} + 3$

Answer

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 3) and other points (7, 2) and (7, negative 2).

9.7-297 §9.7 source ↗

In the following exercises, graph each function using a vertical shift.

$g (x) = x^{2} + 2$

Answer

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 2) and other points (negative 2, 6) and (2, 6).

9.7-299 §9.7 source ↗

In the following exercises, graph each function using a vertical shift.

$h (x) = x^{2} - 4$

Answer

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, negative 4) and other points (negative 2, 0) and (2, 0).

9.7-301 §9.7 source ↗

Graph Quadratic Functions of the form $f (x) = {(x - h)}^{2}$

In the following exercises, ⓐ graph the quadratic functions on the same rectangular coordinate system and ⓑ describe what effect adding a constant, $h$ , inside the parentheses has

$f (x) =$ $x^{2},$ $g (x) =$ ${(x - 3)}^{2},$ and $h (x) =$ ${(x + 3)}^{2} .$

Answer

ⓐ

ⓑ The graph of $g (x) = {(x - 3)}^{2}$ is the same as the graph of $f (x) = x^{2}$ but shifted right 3 units. The graph of $h (x) = {(x + 3)}^{2}$ is the same as the graph of $f (x) = x^{2}$ but shifted left 3 units.

9.7-303 §9.7 source ↗

In the following exercises, graph each function using a horizontal shift.

$f (x) = {(x - 2)}^{2}$

Answer

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (2, 0) and other points (0, 4) and (4, 4).

9.7-305 §9.7 source ↗

In the following exercises, graph each function using a horizontal shift.

$f (x) = {(x + 5)}^{2}$

Answer

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 5, 0) and other points (negative 7, 4) and (negative 3, 4).

9.7-307 §9.7 source ↗

In the following exercises, graph each function using a horizontal shift.

$f (x) = {(x - 5)}^{2}$

Answer

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (5, 0) and other points (3, 4) and (7, 4).

9.7-309 §9.7 source ↗

In the following exercises, graph each function using transformations.

$f (x) =$ ${(x + 2)}^{2} +$ $1$

Answer

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 2, 1) and other points (negative 4, 5) and (0, 5).

9.7-311 §9.7 source ↗

In the following exercises, graph each function using transformations.

$f (x) =$ ${(x - 1)}^{2} +$ $5$

Answer

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (1, 5) and other points (negative 1, 9) and (3, 9).

9.7-313 §9.7 source ↗

In the following exercises, graph each function using transformations.

$f (x) =$ ${(x + 3)}^{2} -$ $1$

Answer

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 3, 1) and other points (negative 4, 0) and (negative 2, 0).

9.7-315 §9.7 source ↗

In the following exercises, graph each function using transformations.

$f (x) =$ ${(x - 4)}^{2} -$ $3$

Answer

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (4, negative 2) and other points (3, negative 2) and (5, negative 2).

9.7-317 §9.7 source ↗

Graph Quadratic Functions of the form $f (x) = a x^{2}$

In the following exercises, graph each function.

$f (x) = −2 x^{2}$

Answer

This figure shows a downward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 1, negative 2) and (1, negative 2).

9.7-319 §9.7 source ↗

Graph Quadratic Functions of the form $f (x) = a x^{2}$

In the following exercises, graph each function.

$f (x) = −4 x^{2}$

Answer

This figure shows a downward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 1, negative 4) and (1, negative 4).

9.7-321 §9.7 source ↗

Graph Quadratic Functions of the form $f (x) = a x^{2}$

In the following exercises, graph each function.

$f (x) = \frac{1}{2} x^{2}$

Answer

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 2, 2) and (2, 2).

9.7-323 §9.7 source ↗

Graph Quadratic Functions of the form $f (x) = a x^{2}$

In the following exercises, graph each function.

$f (x) = \frac{1}{4} x^{2}$

Answer

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (2, 1) and (negative 2, 1).

9.7-325 §9.7 source ↗

Graph Quadratic Functions Using Transformations

In the following exercises, rewrite each function in the $f (x) =$ $a {(x - h)}^{2} +$ $k$ form by completing the square.

$f (x) =$ $−3 x^{2} -$ $12 x -$ $5$

Answer

f (x) =

−3 {(x + 2)}^{2} +

7

9.7-327 §9.7 source ↗

Graph Quadratic Functions Using Transformations

In the following exercises, rewrite each function in the $f (x) =$ $a {(x - h)}^{2} +$ $k$ form by completing the square.

$f (x) =$ $3 x^{2} +$ $6 x -$ $1$

Answer

f (x) =

3 {(x + 1)}^{2} -

4

9.7-329 §9.7 source ↗

In the following exercises, ⓐ rewrite each function in $f (x) =$ $a {(x - h)}^{2} +$ $k$ form and ⓑ graph it by using transformations.

$f (x) =$ $x^{2} +$ $6 x +$ $5$

Answer

ⓐ $f (x) =$ ${(x + 3)}^{2} -$ $4$
ⓑ

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 3, 3), y-intercept of (0, 5), and axis of symmetry shown at x equals negative 3.

9.7-331 §9.7 source ↗

In the following exercises, ⓐ rewrite each function in $f (x) =$ $a {(x - h)}^{2} +$ $k$ form and ⓑ graph it by using transformations.

$f (x) =$ $x^{2} +$ $4 x +$ $3$

Answer

ⓐ $f (x) =$ ${(x + 2)}^{2} -$ $1$
ⓑ

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 2, negative 1), y-intercept of (0, 3), and axis of symmetry shown at x equals negative 2.

9.7-333 §9.7 source ↗

In the following exercises, ⓐ rewrite each function in $f (x) =$ $a {(x - h)}^{2} +$ $k$ form and ⓑ graph it by using transformations.

$f (x) =$ $x^{2} -$ $6 x +$ $15$

Answer

ⓐ $f (x) =$ ${(x - 3)}^{2} +$ $6$
ⓑ

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (3, 6), y-intercept of (0, 10), and axis of symmetry shown at x equals 3.

9.7-335 §9.7 source ↗

In the following exercises, ⓐ rewrite each function in $f (x) =$ $a {(x - h)}^{2} +$ $k$ form and ⓑ graph it by using transformations.

$f (x) =$ $− x^{2} +$ $8 x -$ $16$

Answer

ⓐ $f (x) =$ $− {(x - 4)}^{2} +$ $0$
ⓑ

This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (4, 0), y-intercept of (0, negative 16), and axis of symmetry shown at x equals 4.

9.7-337 §9.7 source ↗

In the following exercises, ⓐ rewrite each function in $f (x) =$ $a {(x - h)}^{2} +$ $k$ form and ⓑ graph it by using transformations.

$f (x) =$ $− x^{2} -$ $4 x +$ $2$

Answer

ⓐ $f (x) =$ $− {(x + 2)}^{2} +$ $6$
ⓑ

This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, 6), y-intercept of (0, 2), and axis of symmetry shown at x equals negative 2.

9.7-339 §9.7 source ↗

In the following exercises, ⓐ rewrite each function in $f (x) =$ $a {(x - h)}^{2} +$ $k$ form and ⓑ graph it by using transformations.

$f (x) =$ $5 x^{2} -$ $10 x +$ $8$

Answer

ⓐ $f (x) =$ $5 {(x - 1)}^{2} +$ $3$
ⓑ

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, 3), y-intercept of (0, 8), and axis of symmetry shown at x equals 1.

9.7-341 §9.7 source ↗

In the following exercises, ⓐ rewrite each function in $f (x) =$ $a {(x - h)}^{2} +$ $k$ form and ⓑ graph it by using transformations.

$f (x) =$ $2 x^{2} -$ $4 x +$ $1$

Answer

ⓐ $f (x) =$ $2 {(x - 1)}^{2} -$ $1$
ⓑ

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 1), y-intercept of (0, 1), and axis of symmetry shown at x equals 1.

9.7-343 §9.7 source ↗

In the following exercises, ⓐ rewrite each function in $f (x) =$ $a {(x - h)}^{2} +$ $k$ form and ⓑ graph it by using transformations.

$f (x) =$ $−2 x^{2} +$ $8 x -$ $10$

Answer

ⓐ $f (x) =$ $−2 {(x - 2)}^{2} -$ $2$
ⓑ

This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (2, negative 2), y-intercept of (0, negative 10), and axis of symmetry shown at x equals 2.

9.7-345 §9.7 source ↗

In the following exercises, ⓐ rewrite each function in $f (x) =$ $a {(x - h)}^{2} +$ $k$ form and ⓑ graph it using properties.

$f (x) =$ $2 x^{2} +$ $4 x +$ $6$

Answer

ⓐ $f (x) =$ $2 {(x + 1)}^{2} +$ $4$
ⓑ

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, 4), y-intercept of (0, 6), and axis of symmetry shown at x equals negative 1.

9.7-347 §9.7 source ↗

In the following exercises, ⓐ rewrite each function in $f (x) =$ $a {(x - h)}^{2} +$ $k$ form and ⓑ graph it using properties.

$f (x) =$ $− x^{2} +$ $2 x -$ $4$

Answer

ⓐ $f (x) =$ $− {(x - 1)}^{2} -$ $3$
ⓑ

This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 3), y-intercept of (0, negative 4), and axis of symmetry shown at x equals 1.

9.7-349 §9.7 source ↗

Matching

In the following exercises, match the graphs to one of the following functions: ⓐ $f (x) = x^{2} + 4$ ⓑ $f (x) = x^{2} - 4$ ⓒ $f (x) = {(x + 4)}^{2}$ ⓓ $f (x) = {(x - 4)}^{2}$ ⓔ $f (x) =$ ${(x + 4)}^{2} -$ $4$ ⓕ $f (x) =$ ${(x + 4)}^{2} +$ $4$ ⓖ $f (x) =$ ${(x - 4)}^{2} -$ $4$ ⓗ $f (x) =$ ${(x - 4)}^{2} +$ $4$

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 4, 0) and other points (negative 4, 4) and (negative 2, 4).

Answer

ⓒ

9.7-351 §9.7 source ↗

Matching

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 4, negative 4) and other points (negative 4, 0) and (negative 2, 0).

Answer

ⓔ

9.7-353 §9.7 source ↗

Matching

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, 0) and other points (2, 4) and (2, 4).

Answer

ⓓ

9.7-355 §9.7 source ↗

Matching

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, negative 4) and other points (2,0) and (6,0).

Answer

ⓖ

9.7-357 §9.7 source ↗

Find a Quadratic Function from its Graph

In the following exercises, write the quadratic function in $f (x) =$ $a {(x - h)}^{2} +$ $k$ form whose graph is shown.

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, negative 5) and y-intercept (0, negative 4).

Answer

f (x) =

{(x + 1)}^{2} -

5

9.7-359 §9.7 source ↗

Find a Quadratic Function from its Graph

In the following exercises, write the quadratic function in $f (x) =$ $a {(x - h)}^{2} +$ $k$ form whose graph is shown.

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 3) and y-intercept (0, negative 1).

Answer

f (x) =

2 {(x - 1)}^{2} -

3

9.7-361 §9.7 source ↗

Find a Quadratic Function from its Graph

In the following exercises, write the quadratic function in $f (x) =$ $a {(x - h)}^{2} +$ $k$ form whose graph is shown.

Graph the quadratic function $f (x) =$ $x^{2} +$ $4 x +$ $5$ first using the properties as we did in the last section and then graph it using transformations. Which method do you prefer? Why?

Answer

Answers will vary.

Be Prepared 9.22 §9.8 source ↗

Solve:

2 x - 3 = 0 .

If you missed this problem, review Example 2.2.

Example 2.2 in the book ↗

Answer

x = \frac{3}{2}

Be Prepared 9.23 §9.8 source ↗

Solve:

2 y^{2} + y = 15

.
If you missed this problem, review Example 6.45.

Example 6.45 in the book ↗

Answer

y = - 3, y = \frac{5}{2}

Be Prepared 9.24 §9.8 source ↗

Solve

\frac{1}{x^{2} + 2 x - 8} >

0

If you missed this problem, review Example 7.56.

Example 7.56 in the book ↗

Answer

- \infty, - 4 \cup 2, \infty

Try It 9.127 §9.8 source ↗

ⓐ Solve

x^{2} +

2 x -

8 <

0

graphically and ⓑ write the solution in interval notation.

Answer

ⓐ

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, negative 9), y-intercept of (0, 8), and axis of symmetry shown at x equals negative 2.

ⓑ $(−4, 2)$

Try It 9.128 §9.8 source ↗

ⓐ Solve

x^{2} - 8 x + 12 \geq 0

graphically and ⓑ write the solution in interval notation.

Answer

ⓐ

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, negative 4) and x-intercepts of (2, 0) and (6, 0).

ⓑ $(− \infty, 2] \cup [6, \infty)$

Try It 9.129 §9.8 source ↗

ⓐ Solve

− x^{2} -

6 x -

5 >

0

graphically and ⓑ write the solution in interval notation.

Answer

ⓐ

A downward-facing parabola on the x y-coordinate plane. It has a vertex of (negative 3, 4), a y-intercept at (0, negative 5), and an axis of symmetry shown at x equals negative 3.

ⓑ $(−5, −1)$

Try It 9.130 §9.8 source ↗

ⓐ Solve

− x^{2} +

10 x -

16 \leq 0

graphically and ⓑ write the solution in interval notation.

Answer

ⓐ

A downward-facing parabola on the x y-coordinate plane. It has a vertex of (5, 9), a y-intercept at (0, negative 16), and an axis of symmetry of x equals 5.

ⓑ $(− \infty, 2] \cup [8, \infty)$

Try It 9.131 §9.8 source ↗

Solve

x^{2} + 2 x - 8 \geq 0

algebraically. Write the solution in interval notation.

Answer

(− \infty, −4] \cup [2, \infty)

Try It 9.132 §9.8 source ↗

Solve

x^{2} - 2 x - 15 \leq 0

algebraically. Write the solution in interval notation.

Answer

[−3, 5]

Try It 9.133 §9.8 source ↗

Solve

− x^{2} + 2 x + 1 \geq 0

algebraically. Write the solution in interval notation.

Answer

[−1 - \sqrt{2}, −1 + \sqrt{2}]

Try It 9.134 §9.8 source ↗

Solve

− x^{2} +

8 x -

14 <

0

algebraically. Write the solution in interval notation.

Answer

(− \infty, 4 - \sqrt{2}) \cup (4 + \sqrt{2}, \infty)

Try It 9.135 §9.8 source ↗

Solve and write any solution in interval notation:
ⓐ

− x^{2} + 2 x - 4 \leq 0

ⓑ

− x^{2} + 2 x - 4 \geq 0

Answer

ⓐ

(− \infty, \infty)

ⓑ no solution

Try It 9.136 §9.8 source ↗

Solve and write any solution in interval notation:
ⓐ

x^{2} +

3 x +

3 <

0

ⓑ

x^{2} +

3 x +

3 >

0

Answer

ⓐ no solution
ⓑ

(− \infty, \infty)

9.8-363 §9.8 source ↗

Solve Quadratic Inequalities Graphically

In the following exercises, ⓐ solve graphically and ⓑ write the solution in interval notation.

$x^{2} +$ $6 x +$ $5 >$ $0$

Answer

ⓐ

The graph shown is an upward-facing parabola with vertex (negative 3, negative 4) and y-intercept (0,5).

ⓑ $(− \infty, −5) \cup (−1, \infty)$

9.8-365 §9.8 source ↗

Solve Quadratic Inequalities Graphically

In the following exercises, ⓐ solve graphically and ⓑ write the solution in interval notation.

$x^{2} + 4 x + 3 \leq 0$

Answer

ⓐ

The graph shown is an upward facing parabola with vertex (negative 2, negative 1) and y-intercept (0,3).

ⓑ $[−3, −1]$

9.8-367 §9.8 source ↗

Solve Quadratic Inequalities Graphically

In the following exercises, ⓐ solve graphically and ⓑ write the solution in interval notation.

$− x^{2} - 3 x + 18 \leq 0$

Answer

ⓐ

The graph shown is a downward-facing parabola with vertex (negative 1 and 5 tenths, 20) and y-intercept (0, 18).

ⓑ $(− \infty, −6] \cup [3, \infty)$

9.8-369 §9.8 source ↗

Solve Quadratic Inequalities Graphically

In the following exercises, ⓐ solve graphically and ⓑ write the solution in interval notation.

$− x^{2} + x + 12 \geq 0$

Answer

ⓐ

The graph shown is a downward facing parabola with a y-intercept of (0, 12) and x-intercepts (negative 3, 0) and (4, 0).

ⓑ $[−3, 4]$

9.8-371 §9.8 source ↗

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

$x^{2} + 3 x - 4 \geq 0$

Answer

(− \infty, −4] \cup [1, \infty)

9.8-373 §9.8 source ↗

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

$x^{2} -$ $7 x +$ $10 <$ $0$

Answer

(2, 5)

9.8-375 §9.8 source ↗

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

$x^{2} +$ $8 x >$ $-$ $15$

Answer

(− \infty, −5) \cup (−3, \infty)

9.8-377 §9.8 source ↗

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

$x^{2} - 4 x + 2 \leq 0$

Answer

[2 - \sqrt{2}, 2 + \sqrt{2}]

9.8-379 §9.8 source ↗

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

$x^{2} -$ $10 x >$ $-$ $19$

Answer

(− \infty, 5 - \sqrt{6}) \cup (5 + \sqrt{6}, \infty)

9.8-381 §9.8 source ↗

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

$−6 x^{2} +$ $19 x -$ $10 \geq 0$

Answer

\frac{2}{3}, \frac{5}{2}

9.8-383 §9.8 source ↗

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

$−2 x^{2} + 7 x + 4 \geq 0$

Answer

[− \frac{1}{2}, 4]

9.8-385 §9.8 source ↗

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

$x^{2} +$ $3 x +$ $5 >$ $0$

Answer

(− \infty, \infty) .

9.8-387 §9.8 source ↗

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

$− x^{2} +$ $x -$ $7 >$ $0$

Answer

no solution

9.8-389 §9.8 source ↗

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

$−2 x^{2} +$ $8 x -$ $10 <$ $0$

Answer

(− \infty, \infty) .

9.8-391 §9.8 source ↗

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

Explain zero partition numbers and how they are used to solve quadratic inequalities algebraically.

Answer

Answers will vary.

9.8-393 §9.8 source ↗

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

Describe the steps needed to solve a quadratic inequality graphically.

Answer

Answers will vary.

9.R-395 §9.R source ↗

Solve Quadratic Equations of the form ax² = k Using the Square Root Property

In the following exercises, solve using the Square Root Property.

$y^{2} = 144$

Answer

y = \pm 12

9.R-397 §9.R source ↗

Solve Quadratic Equations of the form ax² = k Using the Square Root Property

In the following exercises, solve using the Square Root Property.

$4 a^{2} = 100$

Answer

a = \pm 5

9.R-399 §9.R source ↗

Solve Quadratic Equations of the form ax² = k Using the Square Root Property

In the following exercises, solve using the Square Root Property.

$r^{2} + 32 = 0$

Answer

r = \pm 4 \sqrt{2} i

9.R-401 §9.R source ↗

Solve Quadratic Equations of the form ax² = k Using the Square Root Property

In the following exercises, solve using the Square Root Property.

$\frac{2}{3} w^{2} - 20 = 30$

Answer

w = \pm 5 \sqrt{3}

9.R-403 §9.R source ↗

Solve Quadratic Equations of the Form $a {(x - h)}^{2} = k$ Using the Square Root Property

In the following exercises, solve using the Square Root Property.

${(p - 5)}^{2} + 3 = 19$

Answer

p = 1, p = 9

9.R-405 §9.R source ↗

Solve Quadratic Equations of the Form $a {(x - h)}^{2} = k$ Using the Square Root Property

In the following exercises, solve using the Square Root Property.

${(x - \frac{1}{4})}^{2} = \frac{3}{16}$

Answer

x = \frac{1}{4} \pm \frac{\sqrt{3}}{4}

9.R-407 §9.R source ↗

Solve Quadratic Equations of the Form $a {(x - h)}^{2} = k$ Using the Square Root Property

In the following exercises, solve using the Square Root Property.

${(n - 4)}^{2} -$ $50 =$ $150$

Answer

n = 4 \pm 10 \sqrt{2}

9.R-409 §9.R source ↗

Solve Quadratic Equations of the Form $a {(x - h)}^{2} = k$ Using the Square Root Property

In the following exercises, solve using the Square Root Property.

$n^{2} +$ $10 n +$ $25 =$ $12$

Answer

n = −5 \pm 2 \sqrt{3}

9.R-411 §9.R source ↗

Solve Quadratic Equations Using Completing the Square

In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

$x^{2} + 22 x$

Answer

{(x + 11)}^{2}

9.R-413 §9.R source ↗

Solve Quadratic Equations Using Completing the Square

In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

$a^{2} - 3 a$

Answer

{(a - \frac{3}{2})}^{2}

9.R-415 §9.R source ↗

In the following exercises, solve by completing the square.

$d^{2} + 14 d = −13$

Answer

d = −13, −1

9.R-417 §9.R source ↗

In the following exercises, solve by completing the square.

$m^{2} + 6 m = −109$

Answer

m = −3 \pm 10 i

9.R-419 §9.R source ↗

In the following exercises, solve by completing the square.

$v^{2} - 14 v = −31$

Answer

v = 7 \pm 3 \sqrt{2}

9.R-421 §9.R source ↗

In the following exercises, solve by completing the square.

$m^{2} +$ $10 m -$ $4 =$ $−13$

Answer

m = −9, −1

9.R-423 §9.R source ↗

In the following exercises, solve by completing the square.

$a^{2} = 3 a + 8$

Answer

a = \frac{3}{2} \pm \frac{\sqrt{41}}{2}

9.R-425 §9.R source ↗

In the following exercises, solve by completing the square.

$(u + 8) (u + 4) =$ $14$

Answer

u = −6 \pm 3 \sqrt{2}

9.R-427 §9.R source ↗

Solve Quadratic Equations of the form ax² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

$3 p^{2} -$ $18 p +$ $15 =$ $15$

Answer

p = 0, 6

9.R-429 §9.R source ↗

Solve Quadratic Equations of the form ax² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

$4 y^{2} - 6 y = 4$

Answer

y = - \frac{1}{2}, 2

9.R-431 §9.R source ↗

Solve Quadratic Equations of the form ax² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

$3 c^{2} + 2 c = 9$

Answer

c = - \frac{1}{3} \pm \frac{2 \sqrt{7}}{3}

9.R-433 §9.R source ↗

Solve Quadratic Equations of the form ax² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

$2 x^{2} + 6 x = −5$

Answer

x = \frac{3}{2} \pm \frac{1}{2} i

9.R-435 §9.R source ↗

In the following exercises, solve by using the Quadratic Formula.

$4 x^{2} - 5 x + 1 = 0$

Answer

x = \frac{1}{4}, 1

9.R-437 §9.R source ↗

In the following exercises, solve by using the Quadratic Formula.

$r^{2} - r - 42 = 0$

Answer

r = −6, 7

9.R-439 §9.R source ↗

In the following exercises, solve by using the Quadratic Formula.

$4 v^{2} + v - 5 = 0$

Answer

v = - \frac{5}{4}, 1

9.R-441 §9.R source ↗

In the following exercises, solve by using the Quadratic Formula.

$3 m^{2} + 8 m + 2 = 0$

Answer

m = \frac{−4 \pm \sqrt{10}}{3}

9.R-443 §9.R source ↗

In the following exercises, solve by using the Quadratic Formula.

$6 a^{2} - 5 a + 2 = 0$

Answer

a = \frac{5}{12} \pm \frac{\sqrt{23}}{12} i

9.R-445 §9.R source ↗

In the following exercises, solve by using the Quadratic Formula.

$u (u - 10) + 3 = 0$

Answer

u = 5 \pm \sqrt{22}

9.R-447 §9.R source ↗

In the following exercises, solve by using the Quadratic Formula.

$\frac{1}{8} p^{2} -$ $\frac{1}{5} p =$ $-$ $\frac{1}{20}$

Answer

p = \frac{4 \pm \sqrt{6}}{5}

9.R-449 §9.R source ↗

In the following exercises, solve by using the Quadratic Formula.

$4 c^{2} + 4 c + 1 = 0$

Answer

c = - \frac{1}{2}

9.R-451 §9.R source ↗

Use the Discriminant to Predict the Number of Solutions of a Quadratic Equation

In the following exercises, determine the number of solutions for each quadratic equation.

Answer

9.R-453 §9.R source ↗

Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.

Answer

9.R-455 §9.R source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

$x^{4} -$ $14 x^{2} +$ $24 =$ $0$

Answer

x = \pm \sqrt{2}, \pm 2 \sqrt{3}

9.R-457 §9.R source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

$4 x^{4} - 5 x^{2} + 1 = 0$

Answer

x = \pm 1, \pm \frac{1}{2}

9.R-459 §9.R source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

$x + 3 \sqrt{x} - 28 = 0$

Answer

x = 16

9.R-461 §9.R source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

$x^{\frac{2}{3}} -$ $10 x^{\frac{1}{3}} +$ $24 =$ $0$

Answer

x = 64, 216

9.R-463 §9.R source ↗

Solve Equations in Quadratic Form

In the following exercises, solve.

$8 x^{−2} -$ $2 x^{−1} -$ $3 =$ $0$

Answer

x = −2, \frac{4}{3}

9.R-465 §9.R source ↗

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve by using the method of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth, if needed.

Find two consecutive even numbers whose product is 624.

Answer

Two consecutive even numbers whose product is 624 are 24 and 26, and −24 and −26.

9.R-467 §9.R source ↗

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve by using the method of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth, if needed.

Julius built a triangular display case for his coin collection. The height of the display case is six inches less than twice the width of the base. The area of the of the back of the case is 70 square inches. Find the height and width of the case.

Answer

The height is 14 inches and the width is 10 inches.

9.R-469 §9.R source ↗

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve by using the method of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth, if needed.

A rectangular piece of plywood has a diagonal which measures two feet more than the width. The length of the plywood is twice the width. What is the length of the plywood’s diagonal? Round to the nearest tenth.

Answer

The length of the diagonal is 3.6 feet.

9.R-471 §9.R source ↗

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve by using the method of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth, if needed.

For Sophia’s graduation party, several tables of the same width will be arranged end to end to give serving table with a total area of 75 square feet. The total length of the tables will be two more than three times the width. Find the length and width of the serving table so Sophia can purchase the correct size tablecloth . Round answer to the nearest tenth.

Answer

The width of the serving table is 4.7 feet and the length is 16.1 feet.

9.R-473 §9.R source ↗

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve by using the method of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth, if needed.

The couple took a small airplane for a quick flight up to the wine country for a romantic dinner and then returned home. The plane flew a total of 5 hours and each way the trip was 360 miles. If the plane was flying at 150 mph, what was the speed of the wind that affected the plane?

Answer

The speed of the wind was 30 mph.

9.R-475 §9.R source ↗

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve by using the method of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth, if needed.

Two handymen can do a home repair in 2 hours if they work together. One of the men takes 3 hours more than the other man to finish the job by himself. How long does it take for each handyman to do the home repair individually?

Answer

One man takes 3 hours and the other man 6 hours to finish the repair alone.

9.R-477 §9.R source ↗

Recognize the Graph of a Quadratic Function

In the following exercises, graph by plotting point.

Graph $y = − x^{2} + 3$

Answer

This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (3, 0) and other points of (negative 2, negative 1) and (2, negative 1).

9.R-479 §9.R source ↗

In the following exercises, determine if the following parabolas open up or down.

ⓐ $y = x^{2} + 8 x - 1$
ⓑ $y = −4 x^{2} - 7 x + 1$

Answer

ⓐ up ⓑ down

9.R-481 §9.R source ↗

Find the Axis of Symmetry and Vertex of a Parabola

In the following exercises, find ⓐ the equation of the axis of symmetry and ⓑ the vertex.

$y = 2 x^{2} - 8 x + 1$

Answer

x = 2; (2, −7)

9.R-483 §9.R source ↗

Find the Intercepts of a Parabola

In the following exercises, find the x- and y-intercepts.

$y = x^{2} - 8 x + 15$

Answer

\begin{matrix} y : (0, 15) \\ x : (3, 0), (5, 0) \end{matrix}

9.R-485 §9.R source ↗

Find the Intercepts of a Parabola

In the following exercises, find the x- and y-intercepts.

$y =$ $−5 x^{2} -$ $30 x -$ $46$

Answer

\begin{matrix} y : (0, −46) \\ x : none \end{matrix}

9.R-487 §9.R source ↗

Find the Intercepts of a Parabola

In the following exercises, find the x- and y-intercepts.

$y = x^{2} + 16 x + 64$

Answer

\begin{matrix} y : (0, 64) \\ x : (−8, 0) \end{matrix}

9.R-489 §9.R source ↗

Graph Quadratic Functions Using Properties

In the following exercises, graph by using its properties.

$y = x^{2} - 2 x - 3$

Answer

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 4) and a y-intercept of (0, negative 3).

9.R-491 §9.R source ↗

Graph Quadratic Functions Using Properties

In the following exercises, graph by using its properties.

$y = 4 x^{2} - 4 x + 1$

Answer

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (one-half, 0) and a y-intercept of (0, 1).

9.R-493 §9.R source ↗

Graph Quadratic Functions Using Properties

In the following exercises, graph by using its properties.

$y =$ $−2 x^{2} -$ $8 x -$ $12$

Answer

This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, negative 4) and a y-intercept of (0, negative 12).

9.R-495 §9.R source ↗

Solve Maximum and Minimum Applications

In the following exercises, find the minimum or maximum value.

$y =$ $−3 x^{2} +$ $12 x -$ $10$

Answer

The maximum value is 2 when x = 2.

9.R-497 §9.R source ↗

In the following exercises, solve. Rounding answers to the nearest tenth.

A daycare facility is enclosing a rectangular area along the side of their building for the children to play outdoors. They need to maximize the area using 180 feet of fencing on three sides of the yard. The quadratic equation $A (x) = x 90 - \frac{x}{2}$ gives the area, A, of the yard for the length, x, of the building that will border the yard. Find the length of the building that should border the yard to maximize the area, and then find the maximum area.

Answer

The length adjacent to the building is 90 feet giving a maximum area of 4,050 square feet.

9.R-499 §9.R source ↗

Graph Quadratic Functions of the form $f (x) = x^{2} + k$

In the following exercises, graph each function using a vertical shift.

$h (x) = x^{2} - 3$

Answer

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 3, 0) and other points of (negative 1, negative 2) and (1, negative 2).

9.R-501 §9.R source ↗

In the following exercises, graph each function using a horizontal shift.

$g (x) = {(x - 3)}^{2}$

Answer

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (3, 0) and other points of (2, 1) and (4,1).

9.R-503 §9.R source ↗

In the following exercises, graph each function using transformations.

$f (x) =$ ${(x + 3)}^{2} -$ $2$

Answer

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 3, negative 2) and other points of (negative 5, 2) and (negative 1, 2).

9.R-505 §9.R source ↗

In the following exercises, graph each function using transformations.

$f (x) =$ ${(x - 4)}^{2} -$ $3$

Answer

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, negative 3) and other points of (3, negative 2) and (5, negative 2).

9.R-507 §9.R source ↗

Graph Quadratic Functions of the form $f (x) = a x^{2}$

In the following exercises, graph each function.

$f (x) = − x^{2}$

Answer

This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (0, 0) and other points of (negative 1, negative 1) and (1, negative 1).

9.R-509 §9.R source ↗

Graph Quadratic Functions Using Transformations

In the following exercises, rewrite each function in the $f (x) =$ $a {(x - h)}^{2} +$ $k$ form by completing the square.

$f (x) =$ $2 x^{2} -$ $4 x -$ $4$

Answer

f (x) =

2 {(x - 1)}^{2} -

6

9.R-511 §9.R source ↗

In the following exercises, ⓐ rewrite each function in $f (x) =$ $a {(x - h)}^{2} +$ $k$ form and ⓑ graph it by using transformations.

$f (x) =$ $3 x^{2} -$ $6 x -$ $1$

Answer

ⓐ $f (x) =$ $3 {(x - 1)}^{2} -$ $4$
ⓑ

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 4) and other points of (0, negative 1) and (2, negative 1).

9.R-513 §9.R source ↗

In the following exercises, ⓐ rewrite each function in $f (x) =$ $a {(x - h)}^{2} +$ $k$ form and ⓑ graph it by using transformations.

$f (x) =$ $2 x^{2} +$ $4 x +$ $6$

Answer

ⓐ $f (x) =$ $2 {(x + 1)}^{2} +$ $4$
ⓑ

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, 4) and other points of (negative 2, 6) and (0, 6).

9.R-515 §9.R source ↗

In the following exercises, ⓐ rewrite each function in $f (x) =$ $a {(x - h)}^{2} +$ $k$ form and ⓑ graph it using properties.

$f (x) =$ $−3 x^{2} -$ $12 x -$ $5$

Answer

ⓐ $f (x) =$ $−3 {(x + 2)}^{2} +$ $7$
ⓑ

This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, 7) and other points of (negative 4, negative 5) and (0, negative 5).

9.R-517 §9.R source ↗

Find a Quadratic Function from its Graph

In the following exercises, write the quadratic function in $f (x) =$ $a {(x - h)}^{2} +$ $k$ form.

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, negative 1) and other points of (negative 2, negative 4) and (0, negative 4).

Answer

f (x) =

{(x + 1)}^{2} -

5

9.R-519 §9.R source ↗

Solve Quadratic Inequalities Graphically

In the following exercises, solve graphically and write the solution in interval notation.

$x^{2} - x - 6 > 0$

Answer

ⓐ

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (one-half, negative 6 and one-fourth) and other points of (0, negative 6) and (1, negative 6).

ⓑ $(- \infty, −2) \cup (3, \infty)$

9.R-521 §9.R source ↗

Solve Quadratic Inequalities Graphically

In the following exercises, solve graphically and write the solution in interval notation.

$− x^{2} - x + 2 \geq 0$

Answer

ⓐ

This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative one-half, 2 and one-fourth) and other points of (negative 2, 0) and (1, 0).

ⓑ $[−2, 1]$

9.R-523 §9.R source ↗

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

$x^{2} -$ $6 x +$ $8 <$ $0$

Answer

(2, 4)

9.R-525 §9.R source ↗

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

$x^{2} - 6 x + 4 \leq 0$

Answer

[3 - \sqrt{5}, 3 + \sqrt{5}]

9.R-527 §9.R source ↗

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

$− x^{2} +$ $x -$ $6 >$ $0$

Answer

no solution

9.T-529 §9.T source ↗

Use the Square Root Property to solve the quadratic equation

3 {(w + 5)}^{2} = 27 .

Answer

w = −2, w = −8

9.T-531 §9.T source ↗

Use the Quadratic Formula to solve the quadratic equation

2 m^{2} - 5 m + 3 = 0 .

Answer

m = 1, m = \frac{3}{2}

9.T-533 §9.T source ↗

Solve the following quadratic equations. Use any method.

$\frac{9}{4} y^{2} - 3 y + 1 = 0$

Answer

y = \frac{2}{3}

9.T-535 §9.T source ↗

Use the discriminant to determine the number and type of solutions of each quadratic equation.

$3 q^{2} -$ $10 q +$ $12 =$ $0$

Answer

2 complex

9.T-537 §9.T source ↗

Solve each equation.

$y^{\frac{2}{3}} +$ $2 y^{\frac{1}{3}} -$ $3 =$ $0$

Answer

y = 1, y = −27

9.T-539 §9.T source ↗

For each parabola, find ⓐ which direction it opens, ⓑ the equation of the axis of symmetry, ⓒ the vertex, ⓓ the x- and y-intercepts, and e) the maximum or minimum value.

$y = - x^{2} - 8 x - 16$

Answer

ⓐ down ⓑ

x = −4

ⓒ

(−4, 0)

ⓓ

y : (0, −16); x : (−4, 0)

ⓔ maximum value of

0

when

x = −4

9.T-541 §9.T source ↗

Graph each quadratic function using intercepts, the vertex, and the equation of the axis of symmetry.

$f (x) =$ $−2 x^{2} +$ $8 x +$ $4$

Answer

This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (2, 12) and other points of (0, 4) and (4, 4).

9.T-543 §9.T source ↗

In the following exercises, graph each function using transformations.

$f (x) =$ $x^{2} -$ $4 x -$ $1$

Answer

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (2, negative 5) and other points of (0, negative 1) and (4, negative 1).

$f (x) =$ $2 {(x - 1)}^{2} -$ $6$

9.T-545 §9.T source ↗

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

$2 x^{2} +$ $x -$ $10 >$ $0$

Answer

(− \infty, - \frac{5}{2}) \cup (2, \infty)

9.T-547 §9.T source ↗

Model the situation with a quadratic equation and solve by any method.

The length of a diagonal of a rectangle is three more than the width. The length of the rectangle is three times the width. Find the length of the diagonal. (Round to the nearest tenth.)

Answer

The diagonal is 4.4 units long.

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Problems and answers are adapted from Intermediate Algebra 2e by Lynn Marecek and Andrea Honeycutt Mathis, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.