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# 9.4E: Exercises

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### Practice Makes Perfect

##### ExerciseS 1 - 32: Solve Quadratic Equations Using the Quadratic Formula

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

1. $$4 m^{2}+m-3=0$$

2. $$4 n^{2}-9 n+5=0$$

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

4. $$3 q^{2}+8 q-3=0$$

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

6. $$q^{2}+3 q-18=0$$

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

8. $$t^{2}+13 t=-40$$

9. $$3 u^{2}+7 u-2=0$$

10. $$2 p^{2}+8 p+5=0$$

11. $$2 a^{2}-6 a+3=0$$

12. $$5 b^{2}+2 b-4=0$$

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

14. $$y^{2}+4 y-4=0$$

15. $$3 y^{2}+5 y-2=0$$

16. $$6 x^{2}+2 x-20=0$$

17. $$2 x^{2}+3 x+3=0$$

18. $$2 x^{2}-x+1=0$$

19. $$8 x^{2}-6 x+2=0$$

20. $$8 x^{2}-4 x+1=0$$

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

22. $$(x+1)(x-3)=2$$

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

24. $$(x+2)(x+6)=21$$

25. $$\dfrac{1}{4} m^{2}+\dfrac{1}{12} m=\dfrac{1}{3}$$

26. $$\dfrac{1}{3} n^{2}+n=-\dfrac{1}{2}$$

27. $$\dfrac{3}{4} b^{2}+\dfrac{1}{2} b=\dfrac{3}{8}$$

28. $$\dfrac{1}{9} c^{2}+\dfrac{2}{3} c=3$$

29. $$16 c^{2}+24 c+9=0$$

30. $$25 d^{2}-60 d+36=0$$

31. $$25 q^{2}+30 q+9=0$$

32. $$16 y^{2}+8 y+1=0$$

Answer

1. $$m=-1, m=\dfrac{3}{4}$$

3. $$p=\dfrac{1}{3}, p=2$$

5. $$p=-4, p=-3$$

7. $$r=-3, r=11$$

9. $$u=\dfrac{-7 \pm \sqrt{73}}{6}$$

11. $$a=\dfrac{3 \pm \sqrt{3}}{2}$$

13. $$x=-4 \pm 2 \sqrt{5}$$

15. $$y=-\dfrac{2}{3}, y=-1$$

17. $$x=-\dfrac{3}{4} \pm \dfrac{\sqrt{15}}{4} i$$

19. $$x=\dfrac{3}{8} \pm \dfrac{\sqrt{7}}{8} i$$

21. $$v=2 \pm 2 \sqrt{2}$$

23. $$y=-4, y=7$$

25. $$m=1, m=\dfrac{-4}{3}$$

27. $$b=\dfrac{-2 \pm \sqrt{22}}{6}$$

29. $$c=-\dfrac{3}{4}$$

31. $$q=-\dfrac{3}{5}$$

##### ExerciseS 33 - 36 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.

1. $$4 x^{2}-5 x+16=0$$
2. $$36 y^{2}+36 y+9=0$$
3. $$6 m^{2}+3 m-5=0$$
1. $$9 v^{2}-15 v+25=0$$
2. $$100 w^{2}+60 w+9=0$$
3. $$5 c^{2}+7 c-10=0$$
1. $$r^{2}+12 r+36=0$$
2. $$8 t^{2}-11 t+5=0$$
3. $$3 v^{2}-5 v-1=0$$
1. $$25 p^{2}+10 p+1=0$$
2. $$7 q^{2}-3 q-6=0$$
3. $$7 y^{2}+2 y+8=0$$
Answer

33. a. no real solutions b. $$1$$ c. $$2$$

35. a. $$1$$ b. no real solutions c. $$2$$

##### ExerciseS 37 - 40: 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.

1. $$x^{2}-5 x-24=0$$
2. $$(y+5)^{2}=12$$
3. $$14 m^{2}+3 m=11$$
1. $$(8 v+3)^{2}=81$$
2. $$w^{2}-9 w-22=0$$
3. $$4 n^{2}-10=6$$
1. $$6 a^{2}+14=20$$
2. $$\left(x-\dfrac{1}{4}\right)^{2}=\dfrac{5}{16}$$
3. $$y^{2}-2 y=8$$
1. $$8 b^{2}+15 b=4$$
2. $$\dfrac{5}{9} v^{2}-\dfrac{2}{3} v=1$$
3. $$\left(w+\dfrac{4}{3}\right)^{2}=\dfrac{2}{9}$$
Answer

37. a. Factor b. Square Root c. Quadratic Formula

39. a. Quadratic Formula b. Square Root c. Factor

##### ExerciseS 41 - 42: Writing Exercises
1. Solve the equation $$x^{2}+10 x=120$$
1. by completing the square
2. using the Quadratic Formula
3. Which method do you prefer? Why?
2. Solve the equation $$12 y^{2}+23 y=24$$
1. by completing the square
2. using the Quadratic Formula
3. Which method do you prefer? Why?
Answer

41. Answers will vary

## Self Check

a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

b. What does this checklist tell you about your mastery of this section? What steps will you take to improve?

9.4E: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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