Chapter 2: Equations and InequalitiesContent check — College Algebra, Chapter 2: Equations and Inequalities

For each of the following exercises, find the x-intercept and the y-intercept without graphing. Write the coordinates of each intercept.

$y=\mathrm{-3}x+6$

For each of the following exercises, find the x-intercept and the y-intercept without graphing. Write the coordinates of each intercept.

$3x-2y=6$

For each of the following exercises, find the x-intercept and the y-intercept without graphing. Write the coordinates of each intercept.

$3x+8y=9$

For each of the following exercises, solve the equation for y in terms of x.

$4x+2y=8$

For each of the following exercises, solve the equation for y in terms of x.

$2x=5-3y$

For each of the following exercises, solve the equation for y in terms of x.

$5y+4=10x$

For each of the following exercises, find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers.

$(\mathrm{-4},1)$ and $(3,\mathrm{-4})$

For each of the following exercises, find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers.

$(5,0)$ and $(5,6)$

For each of the following exercises, find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers.

Find the distance between the two points given using your calculator, and round your answer to the nearest hundredth.

$(19,12)$ and $(41,71)$

For each of the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.

$(\mathrm{-1},1)$ and $(7,\mathrm{-4})$

For each of the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.

$(0,7)$ and $(4,\mathrm{-9})$

For each of the following exercises, identify the information requested.

What are the coordinates of the origin?

For each of the following exercises, identify the information requested.

If a point is located on the x-axis, what is the y-coordinate?

For each of the following exercises, plot the three points on the given coordinate plane. State whether the three points you plotted appear to be collinear (on the same line).

$(\mathrm{-1},2)(0,4)(2,1)$

For each of the following exercises, plot the three points on the given coordinate plane. State whether the three points you plotted appear to be collinear (on the same line).

Name the coordinates of the points graphed.

For each of the following exercises, construct a table and graph the equation by plotting at least three points.

$y=\frac{1}{3}x+2$

For each of the following exercises, construct a table and graph the equation by plotting at least three points.

$2y=x+3$

For each of the following exercises, find and plot the x- and y-intercepts, and graph the straight line based on those two points.

$x-2y=8$

For each of the following exercises, find and plot the x- and y-intercepts, and graph the straight line based on those two points.

$3y=\mathrm{-2}x+6$

For each of the following exercises, use the graph in the figure below.

Find the distance between the two endpoints using the distance formula. Round to three decimal places.

For each of the following exercises, use the graph in the figure below.

Find the distance that $\left(\mathrm{-3},4\right)$ is from the origin.

For each of the following exercises, use the graph in the figure below.

Which point is closer to the origin?

For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.

After graphing it, use the 2^nd CALC button and 1:value button, hit enter. At the lower part of the screen you will see “x=” and a blinking cursor. You may enter any number for x and it will display the y value for any x value you input. Use this and plug in x = 0, thus finding the y-intercept, for each of the following graphs.

${\text{Y}}_1=\frac{3x-8}{4}$

For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.

After graphing it, use the 2^nd CALC button and 2:zero button, hit ENTER. At the lower part of the screen you will see “left bound?” and a blinking cursor on the graph of the line. Move this cursor to the left of the x-intercept, hit ENTER. Now it says “right bound?” Move the cursor to the right of the x-intercept, hit ENTER. Now it says “guess?” Move your cursor to the left somewhere in between the left and right bound near the x-intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the x-intercept or the “zero” to the y-value. Use this to find the x-intercept.

Note: With linear/straight line functions the zero is not really a “guess,” but it is necessary to enter a “guess” so it will search and find the exact x-intercept between your right and left boundaries. With other types of functions (more than one x-intercept), they may be irrational numbers so “guess” is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.

${\text{Y}}_1=\mathrm{-8}x+6$

For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.

After graphing it, use the 2^nd CALC button and 2:zero button, hit ENTER. At the lower part of the screen you will see “left bound?” and a blinking cursor on the graph of the line. Move this cursor to the left of the x-intercept, hit ENTER. Now it says “right bound?” Move the cursor to the right of the x-intercept, hit ENTER. Now it says “guess?” Move your cursor to the left somewhere in between the left and right bound near the x-intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the x-intercept or the “zero” to the y-value. Use this to find the x-intercept.

Note: With linear/straight line functions the zero is not really a “guess,” but it is necessary to enter a “guess” so it will search and find the exact x-intercept between your right and left boundaries. With other types of functions (more than one x-intercept), they may be irrational numbers so “guess” is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.

${\text{Y}}_1=\frac{3x+5}{4}$ Round your answer to the nearest thousandth.

For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.

After graphing it, use the 2^nd CALC button and 2:zero button, hit ENTER. At the lower part of the screen you will see “left bound?” and a blinking cursor on the graph of the line. Move this cursor to the left of the x-intercept, hit ENTER. Now it says “right bound?” Move the cursor to the right of the x-intercept, hit ENTER. Now it says “guess?” Move your cursor to the left somewhere in between the left and right bound near the x-intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the x-intercept or the “zero” to the y-value. Use this to find the x-intercept.

Note: With linear/straight line functions the zero is not really a “guess,” but it is necessary to enter a “guess” so it will search and find the exact x-intercept between your right and left boundaries. With other types of functions (more than one x-intercept), they may be irrational numbers so “guess” is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.

If the road was made in the previous exercise, how much shorter would the person’s one-way trip be every day?

For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.

After graphing it, use the 2^nd CALC button and 2:zero button, hit ENTER. At the lower part of the screen you will see “left bound?” and a blinking cursor on the graph of the line. Move this cursor to the left of the x-intercept, hit ENTER. Now it says “right bound?” Move the cursor to the right of the x-intercept, hit ENTER. Now it says “guess?” Move your cursor to the left somewhere in between the left and right bound near the x-intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the x-intercept or the “zero” to the y-value. Use this to find the x-intercept.

Note: With linear/straight line functions the zero is not really a “guess,” but it is necessary to enter a “guess” so it will search and find the exact x-intercept between your right and left boundaries. With other types of functions (more than one x-intercept), they may be irrational numbers so “guess” is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.

After finding the two midpoints in the previous exercise, find the distance between the two midpoints to the nearest thousandth.

For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.

After graphing it, use the 2^nd CALC button and 2:zero button, hit ENTER. At the lower part of the screen you will see “left bound?” and a blinking cursor on the graph of the line. Move this cursor to the left of the x-intercept, hit ENTER. Now it says “right bound?” Move the cursor to the right of the x-intercept, hit ENTER. Now it says “guess?” Move your cursor to the left somewhere in between the left and right bound near the x-intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the x-intercept or the “zero” to the y-value. Use this to find the x-intercept.

Note: With linear/straight line functions the zero is not really a “guess,” but it is necessary to enter a “guess” so it will search and find the exact x-intercept between your right and left boundaries. With other types of functions (more than one x-intercept), they may be irrational numbers so “guess” is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.

In the previous exercise, find the coordinates of the midpoint for each diagonal.

For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.

After graphing it, use the 2^nd CALC button and 2:zero button, hit ENTER. At the lower part of the screen you will see “left bound?” and a blinking cursor on the graph of the line. Move this cursor to the left of the x-intercept, hit ENTER. Now it says “right bound?” Move the cursor to the right of the x-intercept, hit ENTER. Now it says “guess?” Move your cursor to the left somewhere in between the left and right bound near the x-intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the x-intercept or the “zero” to the y-value. Use this to find the x-intercept.

Note: With linear/straight line functions the zero is not really a “guess,” but it is necessary to enter a “guess” so it will search and find the exact x-intercept between your right and left boundaries. With other types of functions (more than one x-intercept), they may be irrational numbers so “guess” is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.

If San Jose’s coordinates are $(76,\mathrm{–12})$, where the coordinates represent miles, find the distance between San Jose and San Francisco to the nearest mile.

For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.

After graphing it, use the 2^nd CALC button and 2:zero button, hit ENTER. At the lower part of the screen you will see “left bound?” and a blinking cursor on the graph of the line. Move this cursor to the left of the x-intercept, hit ENTER. Now it says “right bound?” Move the cursor to the right of the x-intercept, hit ENTER. Now it says “guess?” Move your cursor to the left somewhere in between the left and right bound near the x-intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the x-intercept or the “zero” to the y-value. Use this to find the x-intercept.

Note: With linear/straight line functions the zero is not really a “guess,” but it is necessary to enter a “guess” so it will search and find the exact x-intercept between your right and left boundaries. With other types of functions (more than one x-intercept), they may be irrational numbers so “guess” is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.

A person on the top of a building wants to have a guy wire extend to a point on the ground 20 ft from the building. To the nearest foot, how long will the wire have to be if the building is 50 ft tall?

When solving the following equation:

$\frac{2}{x-5}=\frac{4}{x+1}$

explain why we must exclude $x=5$ and $x=\mathrm{-1}$ as possible solutions from the solution set.

For the following exercises, solve the equation for $x.$

$4x-3=5$

For the following exercises, solve the equation for $x.$

$12-$ $5(x+3)=$ $2x-$ $5$

For the following exercises, solve the equation for $x.$

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

For the following exercises, solve the equation for $x.$

$3(2x-1)+$ $x=$ $5x+$ $3$

For the following exercises, solve the equation for $x.$

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

For the following exercises, solve each rational equation for $x.$ State all x-values that are excluded from the solution set.

$2-$ $\frac{3}{x+4}=$ $\frac{x+2}{x+4}$

For the following exercises, solve each rational equation for $x.$ State all x-values that are excluded from the solution set.

$\frac{3x}{x-1}+$ $2=$ $\frac{3}{x-1}$

For the following exercises, solve each rational equation for $x.$ State all x-values that are excluded from the solution set.

$\frac{1}{x}=\frac{1}{5}+\frac{3}{2x}$

For the following exercises, find the equation of the line using the point-slope formula. Write all the final equations using the slope-intercept form.

$\left(1,2\right)$ with a slope of $-\frac{4}{5}$

For the following exercises, find the equation of the line using the point-slope formula. Write all the final equations using the slope-intercept form.

y-intercept is 2, and $\left(4,\mathrm{-1}\right)$

For the following exercises, find the equation of the line using the point-slope formula. Write all the final equations using the slope-intercept form.

$\left(1,3\right)\text{ and }\left(5,5\right)$

For the following exercises, find the equation of the line using the point-slope formula. Write all the final equations using the slope-intercept form.

perpendicular to $\text{3}y=x-4$ and passes through the point $\left(\mathrm{-2},1\right)$ .

For the following exercises, find the equation of the line using the given information.

$\left(1,7\right)$ and $\left(3,7\right)$

For the following exercises, find the equation of the line using the given information.

The slope equals zero and it passes through the point $\left(1,\mathrm{-4}\right).$

For the following exercises, find the equation of the line using the given information.

$(\mathrm{–1},3)$ and $(4,\mathrm{–5})$

For the following exercises, graph the pair of equations on the same axes, and state whether they are parallel, perpendicular, or neither.

$$\begin{array}{l}3x-2y=5\hfill \\ 6y-9x=6\hfill \end{array}$$

For the following exercises, graph the pair of equations on the same axes, and state whether they are parallel, perpendicular, or neither.

$$\begin{array}{l}x=4\\ y=\mathrm{-3}\end{array}$$

For the following exercises, find the slope of the line that passes through the given points.

$\left(\mathrm{-3},2\right)$ and $\left(4,\mathrm{-7}\right)$

For the following exercises, find the slope of the line that passes through the given points.

$\left(\mathrm{-1},\mathrm{-2}\right)$ and $\left(3,4\right)$

For the following exercises, find the slope of the lines that pass through each pair of points and determine whether the lines are parallel or perpendicular.

$$\begin{array}{l}\left(\mathrm{-1},3\right)\text{ and }\left(5,1\right)\\ \left(\mathrm{-2},3\right)\text{ and }\left(0,9\right)\end{array}$$

For the following exercises, express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as Y1, then adjust the ymin and ymax values for your window to include where the y-intercept occurs. State your ymin and ymax values.

$0.537x-$ $2.19y=$ $100$

For the following exercises, express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as Y1, then adjust the ymin and ymax values for your window to include where the y-intercept occurs. State your ymin and ymax values.

$\frac{200-30y}{x}=70$

For the following exercises, express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as Y1, then adjust the ymin and ymax values for your window to include where the y-intercept occurs. State your ymin and ymax values.

Starting with the standard form of an equation $Ax+By=C$ solve this expression for $y$ in terms of $A,B,C$ and $x$. Then put the expression in slope-intercept form.

For the following exercises, express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as Y1, then adjust the ymin and ymax values for your window to include where the y-intercept occurs. State your ymin and ymax values.

Given that the following coordinates are the vertices of a rectangle, prove that this truly is a rectangle by showing the slopes of the sides that meet are perpendicular.

$(–1,1),$ $(2,0),$ $(3,3)$ and $(0,4)$

For the following exercises, express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as Y1, then adjust the ymin and ymax values for your window to include where the y-intercept occurs. State your ymin and ymax values.

The slope for a wheelchair ramp for a home has to be $\frac{1}{12}.$ If the vertical distance from the ground to the door bottom is 2.5 ft, find the distance the ramp has to extend from the home in order to comply with the needed slope.

For the following exercises, use this scenario: The cost of renting a car is $45/wk plus $0.25/mi traveled during that week. An equation to represent the cost would be $y=45+.25x,$ where $x$ is the number of miles traveled.

What is your cost if you travel 50 mi?

For the following exercises, use this scenario: The cost of renting a car is $45/wk plus $0.25/mi traveled during that week. An equation to represent the cost would be $y=45+.25x,$ where $x$ is the number of miles traveled.

Suppose you have a maximum of $100 to spend for the car rental. What would be the maximum number of miles you could travel?

For the following exercises, use the information to find a linear algebraic equation model to use to answer the question being asked.

Beth and Ann are joking that their combined ages equal Sam’s age. If Beth is twice Ann’s age and Sam is 69 yr old, what are Beth and Ann’s ages?

For the following exercises, use this scenario: Two different telephone carriers offer the following plans that a person is considering. Company A has a monthly fee of $20 and charges of $.05/min for calls. Company B has a monthly fee of $5 and charges $.10/min for calls.

Find the model of the total cost of Company A’s plan, using $m$ for the minutes.

For the following exercises, use this scenario: Two different telephone carriers offer the following plans that a person is considering. Company A has a monthly fee of $20 and charges of $.05/min for calls. Company B has a monthly fee of $5 and charges $.10/min for calls.

Find out how many minutes of calling would make the two plans equal.

For the following exercises, use this scenario: A wireless carrier offers the following plans that a person is considering. The Family Plan: $90 monthly fee, unlimited talk and text on up to 8 lines, and data charges of $40 for each device for up to 2 GB of data per device. The Mobile Share Plan: $120 monthly fee for up to 10 devices, unlimited talk and text for all the lines, and data charges of $35 for each device up to a shared total of 10 GB of data. Use $P$ for the number of devices that need data plans as part of their cost.

Find the model of the total cost of the Family Plan.

For the following exercises, use this scenario: A wireless carrier offers the following plans that a person is considering. The Family Plan: $90 monthly fee, unlimited talk and text on up to 8 lines, and data charges of $40 for each device for up to 2 GB of data per device. The Mobile Share Plan: $120 monthly fee for up to 10 devices, unlimited talk and text for all the lines, and data charges of $35 for each device up to a shared total of 10 GB of data. Use $P$ for the number of devices that need data plans as part of their cost.

Assuming they stay under their data limit, find the number of devices that would make the two plans equal in cost.

For exercises 17 and 18, use this scenario: A retired woman has $50,000 to invest but needs to make $6,000 a year from the interest to meet certain living expenses. One bond investment pays 15% annual interest. The rest of it she wants to put in a CD that pays 7%.

If we let $x$ be the amount the woman invests in the 15% bond, how much will she be able to invest in the CD?

For exercises 17 and 18, use this scenario: A retired woman has $50,000 to invest but needs to make $6,000 a year from the interest to meet certain living expenses. One bond investment pays 15% annual interest. The rest of it she wants to put in a CD that pays 7%.

Two planes fly in opposite directions. One travels 450 mi/h and the other 550 mi/h. How long will it take before they are 4,000 mi apart?

For exercises 17 and 18, use this scenario: A retired woman has $50,000 to invest but needs to make $6,000 a year from the interest to meet certain living expenses. One bond investment pays 15% annual interest. The rest of it she wants to put in a CD that pays 7%.

Fiora starts riding her bike at 20 mi/h. After a while, she slows down to 12 mi/h, and maintains that speed for the rest of the trip. The whole trip of 70 mi takes her 4.5 h. For what distance did she travel at 20 mi/h?

For exercises 17 and 18, use this scenario: A retired woman has $50,000 to invest but needs to make $6,000 a year from the interest to meet certain living expenses. One bond investment pays 15% annual interest. The rest of it she wants to put in a CD that pays 7%.

Raúl has $20,000 to invest. His intent is to earn 11% interest on his investment. He can invest part of his money at 8% interest and part at 12% interest. How much does Raúl need to invest in each option to make get a total 11% return on his $20,000?

For the following exercises, use this scenario: A truck rental agency offers two kinds of plans. Plan A charges $75/wk plus $.10/mi driven. Plan B charges $100/wk plus $.05/mi driven.

Write the model equation for the cost of renting a truck with plan B.

For the following exercises, use this scenario: A truck rental agency offers two kinds of plans. Plan A charges $75/wk plus $.10/mi driven. Plan B charges $100/wk plus $.05/mi driven.

If Tim knows he has to travel 300 mi, which plan should he choose?

For the following exercises, use the formula given to solve for the required value.

The formula $F=\frac{m{v}^2}{R}$ relates force $\left(F\right)$, velocity $\left(v\right)$, mass , and resistance $\left(m\right)$. Find $R$ when $m=45,$ $v=7,$ and $F=245.$

For the following exercises, use the formula given to solve for the required value.

$Sum=\frac{1}{1-r}$ is the formula for an infinite series sum. If the sum is 5, find $r.$

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question.

Use the formula from the previous question to find the width, $W,$ of a rectangle whose length is 15 and whose perimeter is 58.

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question.

Use the formula from the previous question to find $f$ when $p=$ $8\text{and }q=$ $13.$

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question.

Use the formula from the previous question to find $m$ when the coordinates of the point are $\left(4,7\right)$ and $b=12.$

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question.

Solve for h: $A=$ $\frac{1}{2}h(b_1+b_2)$

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question.

Find the dimensions of an American football field. The length is 200 ft more than the width, and the perimeter is 1,040 ft. Find the length and width. Use the perimeter formula $P=2L+2W.$

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question.

Using the formula in the previous exercise, find the distance that Susan travels if she is moving at a rate of 60 mi/h for 6.75 h.

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question.

If the area model for a triangle is $A=\frac{1}{2}bh,$ find the area of a triangle with a height of 16 in. and a base of 11 in.

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question.

Use the formula from the previous question to find the height to the nearest tenth of a triangle with a base of 15 and an area of 215.

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question.

Solve for h: $V=\pi r^{2}h$

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question.

Solve for r: $V=\pi r^{2}h$

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question.

The formula for the circumference of a circle is $C=2\pi r.$ Find the circumference of a circle with a diameter of 12 in. (diameter = 2r). Use the symbol $\pi$ in your final answer.

For the following exercises, evaluate the algebraic expressions.

If $y=x^2+x-4,$ evaluate $y$ given $x=2i.$

For the following exercises, evaluate the algebraic expressions.

If $y=x^2+3x+5,$ evaluate $y$ given $x=2+i.$

For the following exercises, evaluate the algebraic expressions.

If $y=\frac{x+1}{2-x},$ evaluate $y$ given $x=5i.$

For the following exercises, plot the complex numbers on the complex plane.

$1-2i$

For the following exercises, plot the complex numbers on the complex plane.

$i$

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$(3+2i)+$ $(5-3i)$

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$(\mathrm{-5}+3i)-$ $(6-i)$

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$(\mathrm{-4}+4i)-$ $(\mathrm{-6}+9i)$

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$\left(5-2i\right)(3i)$

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$\left(\mathrm{-2}+4i\right)\left(8\right)$

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$\left(\mathrm{-1}+2i\right)(\mathrm{-2}+3i)$

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$\left(3+4i\right)\left(3-4i\right)$

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$\frac{6-2i}{3}$

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$\frac{6+4i}{i}$

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$\frac{3+4i}{2-i}$

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$\sqrt{\mathrm{-9}}+3\sqrt{\mathrm{-16}}$

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$\frac{2+\sqrt{-12}}{2}$

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$i^8$

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$i^{22}$

For the following exercises, use a calculator to help answer the questions.

Evaluate ${(1-i)}^k$ for $k=2,6,$ and $10$. Predict the value if $k=14.$

For the following exercises, use a calculator to help answer the questions.

Show that a solution of $x^6+1=0$ is $\frac{\sqrt{3}}{2}+\frac{1}{2}i.$

For the following exercises, evaluate the expressions, writing the result as a simplified complex number.

$\frac{1}{i}+\frac{4}{i^3}$

For the following exercises, evaluate the expressions, writing the result as a simplified complex number.

$i^7\left(1+i^2\right)$

For the following exercises, evaluate the expressions, writing the result as a simplified complex number.

$\frac{\left(2+i\right)\left(4-2i\right)}{(1+i)}$

For the following exercises, evaluate the expressions, writing the result as a simplified complex number.

$\frac{{\left(3+i\right)}^2}{{\left(1+2i\right)}^2}$

For the following exercises, evaluate the expressions, writing the result as a simplified complex number.

$\frac{4+i}{i}+$ $\frac{3-4i}{1-i}$

For the following exercises, solve the quadratic equation by factoring.

$x^2-9x+18=0$

For the following exercises, solve the quadratic equation by factoring.

$6x^2+17x+5=0$

For the following exercises, solve the quadratic equation by factoring.

$3x^2-75=0$

For the following exercises, solve the quadratic equation by factoring.

$4x^2=9$

For the following exercises, solve the quadratic equation by factoring.

$5x^2=5x+30$

For the following exercises, solve the quadratic equation by factoring.

$7x^2+3x=0$

For the following exercises, solve the quadratic equation by using the square root property.

$x^2=36$

For the following exercises, solve the quadratic equation by using the square root property.

${\left(x-1\right)}^2=25$

For the following exercises, solve the quadratic equation by using the square root property.

${\left(2x+1\right)}^2=9$

For the following exercises, solve the quadratic equation by completing the square. Show each step.

$x^2-9x-22=0$

For the following exercises, solve the quadratic equation by completing the square. Show each step.

$x^2-6x=13$

For the following exercises, solve the quadratic equation by completing the square. Show each step.

$2+z=6z^2$

For the following exercises, solve the quadratic equation by completing the square. Show each step.

$2x^2-3x-1=0$

For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.

$x^2+4x+7=0$

For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.

$9x^2-$ $30x+$ $25=$ $0$

For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.

$6x^2-x-2=0$

For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution.

$x^2+x=4$

For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution.

$3x^2-5x+1=0$

For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution.

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

For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the x-intercepts) by using 2^nd CALC 2:zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth.

${\text{Y}}_1=$ $\mathrm{-3}{x}^2+$ $8x-$ $1$

For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the x-intercepts) by using 2^nd CALC 2:zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth.

To solve the quadratic equation $x^2+5x-7=4,$ we can graph these two equations

$$\begin{array}{l}\hfill \\ \begin{array}{l}{\text{Y}}_1=x^2+5x-7\hfill \\ {\text{Y}}_2=4\hfill \end{array}\hfill \end{array}$$

and find the points of intersection. Recall 2^nd CALC 5:intersection. Do this and find the solutions to the nearest tenth.

For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the x-intercepts) by using 2^nd CALC 2:zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth.

Beginning with the general form of a quadratic equation, $a{x}^2+bx+c=0,$ solve for x by using the completing the square method, thus deriving the quadratic formula.

For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the x-intercepts) by using 2^nd CALC 2:zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth.

A person has a garden that has a length 10 feet longer than the width. Set up a quadratic equation to find the dimensions of the garden if its area is 119 ft.². Solve the quadratic equation to find the length and width.

For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the x-intercepts) by using 2^nd CALC 2:zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth.

Suppose that an equation is given $p=$ $\mathrm{-2}{x}^2+$ $280x-$ $1000,$ where $x$ represents the number of items sold at an auction and $p$ is the profit made by the business that ran the auction. How many items sold would make this profit a maximum? Solve this by graphing the expression in your graphing utility and finding the maximum using 2^nd CALC maximum. To obtain a good window for the curve, set $x$ [0,200] and $y$ [0,10000].

For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the x-intercepts) by using 2^nd CALC 2:zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth.

The cost function for a certain company is $C=60x+300$ and the revenue is given by $R=$ $100x-$ $0.5x^2.$ Recall that profit is revenue minus cost. Set up a quadratic equation and find two values of x (production level) that will create a profit of $300.

For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the x-intercepts) by using 2^nd CALC 2:zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth.

A vacant lot is being converted into a community garden. The garden and the walkway around its perimeter have an area of 378 ft². Find the width of the walkway if the garden is 12 ft. wide by 15 ft. long.