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9.E: Review Exercises and Sample Exam

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Sep 2, 2024
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- 9.6: Introduction to Complex Numbers and Complex Solutions
- 10: Appendix - Geometric Figures

This page is a draft and is under active development.

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Review Exercises

Exercise $\PageIndex{1}$ extracting square roots

Solve by extracting the roots.

$x^{2}−16=0$
$y^{2}=94$
$x^{2}−27=0$
$x^{2}+27=0$
$3y^{2}−25=0$
$9x^{2}−2=0$
$(x−5)^{2}−9=0$
$(2x−1)^{2}−1=0$
$16(x−6)^{2}−3=0$
$2(x+3)^{2}−5=0$
$(x+3)(x−2)=x+12$
$(x+2)(5x−1)=9x−1$

Answer

1. $±16$

3. $±3\sqrt{3}$

5. $±\frac{5 \sqrt{3}}{3}$

7. $2, 8$

9. $6±\frac{\sqrt{3}}{4}$

11. $±3\sqrt{2}$

Exercise $\PageIndex{2}$ extracting square roots

Find a quadratic equation in standard form with the given solutions.

$\pm\sqrt{2}$
$\pm2\sqrt{5}$

Answer: 1. $x^{2}-2=0$

Exercise $\PageIndex{3}$ completing the square

Complete the square.

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

Answer: 1. $x^{2}-6x+9=(x-3)^{2}$

Exercise $\PageIndex{4}$ completing the square

Solve by completing the square.

$x^{2}−12x+1=0$
$x^{2}+8x+3=0$
$y^{2}−4y−14=0$
$y^{2}−2y−74=0$
$x^{2}+5x−1=0$
$x^{2}−7x−2=0$
$2x^{2}+x−3=0$
$5x^{2}+9x−2=0$
$2x^{2}−16x+5=0$
$3x^{2}−6x+1=0$
$2y^{2}+10y+1=0$
$5y^{2}+y−3=0$
$x(x+9)=5x+8$
$(2x+5)(x+2)=8x+7$

Answer

1. $6±\sqrt{35}$

3. $2±3\sqrt{2}$

5. $\frac{-5±\sqrt{29}}{2}$

7. $\frac{−3}{2}, 1$

9. $\frac{8±3\sqrt{6}}{2}$

11. $\frac{-5±\sqrt{23}}{2}$

13. $−2±2\sqrt{3}$

Exercise $\PageIndex{5}$ quadratic formula

Identify the coefficients a, b, and c used in the quadratic formula. Do not solve.

$x^{2}−x+4=0$
$−x^{2}+5x−14=0$
$x^{2}−5=0$
$6x^{2}+x=0$

Answer

1. $a=1, b=−1,$ and $c=4$

3. $a=1, b=0,$ and $c=−5$

Exercise $\PageIndex{6}$ quadratic formula

Use the quadratic formula to solve the following.

$x^{2}−6x+6=0$
$x^{2}+10x+23=0$
$3y^{2}−y−1=0$
$2y^{2}−3y+5=0$
$5x^{2}−36=0$
$7x^{2}+2x=0$
$−x^{2}+5x+1=0$
$−4x^{2}−2x+1=0$
$t^{2}−12t−288=0$
$t^{2}−44t+484=0$
$(x−3)^{2}−2x=47$
$9x(x+1)−5=3x$

Answer

1. $3±\sqrt{3}$

3. $\frac{1±\sqrt{13}}{6}$

5. $±\frac{6\sqrt{5}}{5}$

7. $\frac{5±\sqrt{29}}{2}$

9. $−12, 24$

11. $4±3\sqrt{6}$

Exercise $\PageIndex{7}$ Guidelines for Solving Quadratic Equations and Applications

Use the discriminant to determine the number and type of solutions.

$−x^{2}+5x+1=0$
$−x^{2}+x−1=0$
$4x^{2}−4x+1=0$
$9x^{2}−4=0$

Answer

1. Two real solutions

3. One real solution

Exercise $\PageIndex{8}$ Guidelines for Solving Quadratic Equations and Applications

Solve using any method.

$x^{2}+4x−60=0$
$9x^{2}+7x=0$
$25t^{2}−1=0$
$t^{2}+16=0$
$x^{2}−x−3=0$
$9x^{2}+12x+1=0$
$4(x−1)^{2}−27=0$
$(3x+5)^{2}−4=0$
$(x−2)(x+3)=6$
$x(x−5)=12$
$(x+1)(x−8)+28=3x$
$(9x−2)(x+4)=28x−9$

Answer

1. $−10, 6$

3. $±\frac{1}{5}$

5. $\frac{1±\sqrt{13}}{2}$

7. $1 ± \frac{3 \sqrt{3}}{2}$

9. $−4, 3$

11. $5±\sqrt{5}$

Exercise $\PageIndex{9}$ Guidelines for Solving Quadratic Equations and Applications

Set up an algebraic equation and use it to solve the following.

The length of a rectangle is 2 inches less than twice the width. If the area measures 25 square inches, then find the dimensions of the rectangle. Round off to the nearest hundredth.
An 18-foot ladder leaning against a building reaches a height of 17 feet. How far is the base of the ladder from the wall? Round to the nearest tenth of a foot.
The value in dollars of a new car is modeled by the function $V(t)=125t^{2}−3,000t+22,000$ , where t represents the number of years since it was purchased. Determine the age of the car when its value is $22,000.
The height in feet reached by a baseball tossed upward at a speed of 48 feet/second from the ground is given by the function $h(t)=−16t^{2}+48t$ , where t represents time in seconds. At what time will the baseball reach a height of 16 feet?

Answer

1. Length: 6.14 inches; width: 4.07 inches

3. It is worth $22,000 new and when it is 24 years old.

Exercise $\PageIndex{10}$ graphing parabolas

Determine the x- and y-intercepts.

$y=2x^{2}+5x−3$
$y=x^{2}−12$
$y=5x^{2}−x+2$
$y=−x^{2}+10x−25$

Answer

1. x-intercepts: $(−3, 0), (\frac{1}{2}, 0)$ ; y-intercept: $(0, −3)$

3. x-intercepts: none; y-intercept: $(0, 2)$

Exercise $\PageIndex{11}$ graphing parabolas

Find the vertex and the line of symmetry.

$y=x^{2}−6x+1$
$y=−x^{2}+8x−1$
$y=x^{2}+3x−1$
$y=9x^{2}−1$

Answer

1. Vertex: $(3, −8)$ ; line of symmetry: $x=3$

3. Vertex: $(−\frac{3}{2}, −\frac{13}{4})$ ; line of symmetry: $x=−32$

Exercise $\PageIndex{12}$ graphing parabolas

Graph. Find the vertex and the y-intercept. In addition, find the x-intercepts if they exist.

$y=x^{2}+8x+12$
$y=−x^{2}−6x+7$
$y=−2x^{2}−4$
$y=x^{2}+4x$
$y=4x^{2}−4x+1$
$y=−2x^{2}$
$y=−2x^{2}+8x−7$
$y=3x^{2}−1$

Answer

Exercise $\PageIndex{13}$ graphing parabolas

Determine the maximum or minimum y-value.

$y=x^{2}−10x+1$
$y=−x^{2}+12x−1$
$y=−5x^{2}+6x$
$y=2x^{2}−x−1$
The value in dollars of a new car is modeled by the function $V(t)=125t^{2}−3,000t+22,000$ , where t represents the number of years since it was purchased. Determine the age of the car when its value is at a minimum.
The height in feet reached by a baseball tossed upward at a speed of 48 feet/second from the ground is given by the function $h(t)=−16t^{2}+48t$ , where t represents time in seconds. What is the maximum height of the baseball?

Answer

1. Minimum: $y = −24$

3. Maximum: $y = \frac{9}{5}$

5. The car will have a minimum value 12 years after it is purchased.

Exercise $\PageIndex{14}$ introduction to complex numbers and complex solutions

Rewrite in terms of i.

$\sqrt{−36}$
$\sqrt{−40}$
$\sqrt{−\frac{8}{25}}$
- $\sqrt{−19}$

Answer

1. 6i

3. $\frac{2 \sqrt{2} i}{5}$

Exercise $\PageIndex{15}$ introduction to complex numbers and complex solutions

Perform the operations.

$(2−5i)+(3+4i)$
$(6−7i)−(12−3i)$
$(2−3i)(5+i)$
$4−i^{2}−3i$

Answer

1. $5−i$

3. $13−13i$

Exercise $\PageIndex{16}$ introduction to complex numbers and complex solutions

Solve.

$9x^{2}+25=0$
$3x^{2}+1=0$
$y^{2}−y+5=0$
$y^{2}+2y+4$
$4x(x+2)+5=8x$
$2(x+2)(x+3)=3(x^{2}+13)$

Answer

1. $\pm\frac{3}{3}$

3. $\frac{1}{2}\pm i \frac{\sqrt{19}}{2}$

5. $\pm i \frac{\sqrt{5}}{2}$

Sample Exam

Exercise $\PageIndex{17}$

Solve by extracting the roots.

$4x^{2}−9=0$
$(4x+1)^{2}−5=0$

Answer: 1. $\pm\frac{3}{2}$

Exercise $\PageIndex{18}$

Solve by completing the square.

$x^{2}+10x+19=0$
$x^{2}−x−1=0$

Answer: 1. $-5\pm\sqrt{6}$

Exercise $\PageIndex{19}$

Solve using the quadratic formula.

$−2x^{2}+x+3=0$
$x^{2}+6x−31=0$

Answer: 1. $-1, \frac{3}{2}$

Exercise $\PageIndex{20}$

Solve using any method.

$(5x+1)(x+1)=1$
$(x+5)(x−5)=65$
$x(x+3)=−2$
$2(x−2)^{2}−6=3x^{2}$

Answer

1. $-\frac{6}{5}, 0$

3. $-2, -1$

Exercise $\PageIndex{21}$

Set up an algebraic equation and solve.

The length of a rectangle is twice its width. If the diagonal measures $6\sqrt{5}$ centimeters, then find the dimensions of the rectangle.
The height in feet reached by a model rocket launched from a platform is given by the function $h(t)=−16t^{2}+256t+3$ , where t represents time in seconds after launch. At what time will the rocket reach 451 feet?

Answer: 1. Length: 12 centimeters; width: 6 centimeters

Exercise $\PageIndex{22}$

Graph. Find the vertex and the y-intercept. In addition, find the x-intercepts if they exist.

$y=2x^{2}−4x−6$
$y=−x^{2}+4x−4$
$y=4x^{2}−9$
$y=x^{2}+2x−1$
Determine the maximum or minimum y-value: $y=−3x^{2}+12x−15$ .
Determine the x- and y-intercepts: $y=x^{2}+x+4$ .
Determine the domain and range: $y=25x^{2}−10x+1$ .
The height in feet reached by a model rocket launched from a platform is given by the function $h(t)=−16t^{2}+256t+3$ , where t represents time in seconds after launch. What is the maximum height attained by the rocket.
A bicycle manufacturing company has determined that the weekly revenue in dollars can be modeled by the formula $R=200n−n^{2}$ , where n represents the number of bicycles produced and sold. How many bicycles does the company have to produce and sell in order to maximize revenue?
Rewrite in terms of i: $\sqrt{−60}$ .
Divide: $\frac{4−2i}{4+2i}$ .

Answer

5. Maximum: $y = −3$

7. Domain: R; range: $[0,∞)$

9. To maximize revenue, the company needs to produce and sell 100 bicycles a week.

11. $\frac{3}{5}−i\frac{4}{5}$

Exercise $\PageIndex{23}$

Solve.

$25x^{2}+3=0$
$−2x^{2}+5x−1=0$

Answer: 2. $\frac{5\pm\sqrt{17}}{4}$

Review Exercises

Exercise 9.E.1\PageIndex{1} extracting square roots

Exercise 9.E.2\PageIndex{2} extracting square roots

Exercise 9.E.3\PageIndex{3} completing the square

Exercise 9.E.4\PageIndex{4} completing the square

Exercise 9.E.5\PageIndex{5} quadratic formula

Exercise 9.E.6\PageIndex{6} quadratic formula

Exercise 9.E.7\PageIndex{7} Guidelines for Solving Quadratic Equations and Applications

Exercise 9.E.8\PageIndex{8} Guidelines for Solving Quadratic Equations and Applications

Exercise 9.E.9\PageIndex{9} Guidelines for Solving Quadratic Equations and Applications

Exercise 9.E.10\PageIndex{10} graphing parabolas

Exercise 9.E.11\PageIndex{11} graphing parabolas

Exercise 9.E.12\PageIndex{12} graphing parabolas

Exercise 9.E.13\PageIndex{13} graphing parabolas

Exercise 9.E.14\PageIndex{14} introduction to complex numbers and complex solutions

Exercise 9.E.15\PageIndex{15} introduction to complex numbers and complex solutions

Exercise 9.E.16\PageIndex{16} introduction to complex numbers and complex solutions

Sample Exam

Exercise 9.E.17\PageIndex{17}

Exercise 9.E.18\PageIndex{18}

Exercise 9.E.19\PageIndex{19}

Exercise 9.E.20\PageIndex{20}

Exercise 9.E.21\PageIndex{21}

Exercise 9.E.22\PageIndex{22}

Exercise 9.E.23\PageIndex{23}

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Exercise $\PageIndex{1}$ extracting square roots

Exercise $\PageIndex{2}$ extracting square roots

Exercise $\PageIndex{3}$ completing the square

Exercise $\PageIndex{4}$ completing the square

Exercise $\PageIndex{5}$ quadratic formula

Exercise $\PageIndex{6}$ quadratic formula

Exercise $\PageIndex{7}$ Guidelines for Solving Quadratic Equations and Applications

Exercise $\PageIndex{8}$ Guidelines for Solving Quadratic Equations and Applications

Exercise $\PageIndex{9}$ Guidelines for Solving Quadratic Equations and Applications

Exercise $\PageIndex{10}$ graphing parabolas

Exercise $\PageIndex{11}$ graphing parabolas

Exercise $\PageIndex{12}$ graphing parabolas

Exercise $\PageIndex{13}$ graphing parabolas

Exercise $\PageIndex{14}$ introduction to complex numbers and complex solutions

Exercise $\PageIndex{15}$ introduction to complex numbers and complex solutions

Exercise $\PageIndex{16}$ introduction to complex numbers and complex solutions

Exercise $\PageIndex{17}$

Exercise $\PageIndex{18}$

Exercise $\PageIndex{19}$

Exercise $\PageIndex{20}$

Exercise $\PageIndex{21}$

Exercise $\PageIndex{22}$

Exercise $\PageIndex{23}$