
# 9.E: Review Exercises and Sample Exam

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## Review Exercises

Exercise $$\PageIndex{1}$$ extracting square roots

Solve by extracting the roots.

1. $$x^{2}−16=0$$
2. $$y^{2}=94$$
3. $$x^{2}−27=0$$
4. $$x^{2}+27=0$$
5. $$3y^{2}−25=0$$
6. $$9x^{2}−2=0$$
7. $$(x−5)^{2}−9=0$$
8. $$(2x−1)^{2}−1=0$$
9. $$16(x−6)^{2}−3=0$$
10. $$2(x+3)^{2}−5=0$$
11. $$(x+3)(x−2)=x+12$$
12. $$(x+2)(5x−1)=9x−1$$

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.

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

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

Exercise $$\PageIndex{3}$$ completing the square

Complete the square.

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

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

Exercise $$\PageIndex{4}$$ completing the square

Solve by completing the square.

1. $$x^{2}−12x+1=0$$
2. $$x^{2}+8x+3=0$$
3. $$y^{2}−4y−14=0$$
4. $$y^{2}−2y−74=0$$
5. $$x^{2}+5x−1=0$$
6. $$x^{2}−7x−2=0$$
7. $$2x^{2}+x−3=0$$
8. $$5x^{2}+9x−2=0$$
9. $$2x^{2}−16x+5=0$$
10. $$3x^{2}−6x+1=0$$
11. $$2y^{2}+10y+1=0$$
12. $$5y^{2}+y−3=0$$
13. $$x(x+9)=5x+8$$
14. $$(2x+5)(x+2)=8x+7$$

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.

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

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.

1. $$x^{2}−6x+6=0$$
2. $$x^{2}+10x+23=0$$
3. $$3y^{2}−y−1=0$$
4. $$2y^{2}−3y+5=0$$
5. $$5x^{2}−36=0$$
6. $$7x^{2}+2x=0$$
7. $$−x^{2}+5x+1=0$$
8. $$−4x^{2}−2x+1=0$$
9. $$t^{2}−12t−288=0$$
10. $$t^{2}−44t+484=0$$
11. $$(x−3)^{2}−2x=47$$
12. $$9x(x+1)−5=3x$$

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.

1. $$−x^{2}+5x+1=0$$
2. $$−x^{2}+x−1=0$$
3. $$4x^{2}−4x+1=0$$
4. $$9x^{2}−4=0$$

1. Two real solutions

3. One real solution

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

Solve using any method.

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

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.

1. 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.
2. 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.
3. 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. 4. 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.

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

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.

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

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.

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

1.

3.

5.

7.

Exercise $$\PageIndex{13}$$ graphing parabolas

Determine the maximum or minimum y-value.

1. $$y=x^{2}−10x+1$$
2. $$y=−x^{2}+12x−1$$
3. $$y=−5x^{2}+6x$$
4. $$y=2x^{2}−x−1$$
5. 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.
6. 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?

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.

1. $$\sqrt{−36}$$
2. $$\sqrt{−40}$$
3. $$\sqrt{−\frac{8}{25}}$$
4. -$$\sqrt{−19}$$

1. 6i

3. $$\frac{2 \sqrt{2} i}{5}$$

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

Perform the operations.

1. $$(2−5i)+(3+4i)$$
2. $$(6−7i)−(12−3i)$$
3. $$(2−3i)(5+i)$$
4. $$4−i^{2}−3i$$

1. $$5−i$$

3. $$13−13i$$

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

Solve.

1. $$9x^{2}+25=0$$
2. $$3x^{2}+1=0$$
3. $$y^{2}−y+5=0$$
4. $$y^{2}+2y+4$$
5. $$4x(x+2)+5=8x$$
6. $$2(x+2)(x+3)=3(x^{2}+13)$$

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.

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

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

Exercise $$\PageIndex{18}$$

Solve by completing the square.

1. $$x^{2}+10x+19=0$$
2. $$x^{2}−x−1=0$$

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

Exercise $$\PageIndex{19}$$

1. $$−2x^{2}+x+3=0$$
2. $$x^{2}+6x−31=0$$

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

Exercise $$\PageIndex{20}$$

Solve using any method.

1. $$(5x+1)(x+1)=1$$
2. $$(x+5)(x−5)=65$$
3. $$x(x+3)=−2$$
4. $$2(x−2)^{2}−6=3x^{2}$$

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

3. $$-2, -1$$

Exercise $$\PageIndex{21}$$

Set up an algebraic equation and solve.

1. The length of a rectangle is twice its width. If the diagonal measures $$6\sqrt{5}$$ centimeters, then find the dimensions of the rectangle.
2. 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?

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.

1. $$y=2x^{2}−4x−6$$
2. $$y=−x^{2}+4x−4$$
3. $$y=4x^{2}−9$$
4. $$y=x^{2}+2x−1$$
5. Determine the maximum or minimum y-value: $$y=−3x^{2}+12x−15$$.
6. Determine the x- and y-intercepts: $$y=x^{2}+x+4$$.
7. Determine the domain and range: $$y=25x^{2}−10x+1$$.
8. 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.
9. 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?
10. Rewrite in terms of i: $$\sqrt{−60}$$.
11. Divide: $$\frac{4−2i}{4+2i}$$.

1.

3.

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.

1. $$25x^{2}+3=0$$
2. $$−2x^{2}+5x−1=0$$
2. $$\frac{5\pm\sqrt{17}}{4}$$