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9.3E: Exercises

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Practice Makes Perfect

ExerciseS 1 - 4: 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.

1. $$m^{2}-24 m$$
2. $$x^{2}-11 x$$
3. $$p^{2}-\frac{1}{3} p$$
1. $$n^{2}-16 n$$
2. $$y^{2}+15 y$$
3. $$q^{2}+\frac{3}{4} q$$
1. $$p^{2}-22 p$$
2. $$y^{2}+5 y$$
3. $$m^{2}+\frac{2}{5} m$$
1. $$q^{2}-6 q$$
2. $$x^{2}-7 x$$
3. $$n^{2}-\frac{2}{3} n$$
Answer

1. a. $$(m-12)^{2}$$ b. $$\left(x-\frac{11}{2}\right)^{2}$$ c. $$\left(p-\frac{1}{6}\right)^{2}$$

3. a. $$(p-11)^{2}$$ b. $$\left(y+\frac{5}{2}\right)^{2}$$ c. $$\left(m+\frac{1}{5}\right)^{2}$$

ExerciseS 5 - 28: Solve Quadratic Equations of the Form $$x^{2}+b x+c=0$$ by Completing the Square

In the following exercises, solve by completing the square.

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

6. $$z^{2}+12 z=-11$$

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

8. $$y^{2}-2 y=8$$

9. $$m^{2}+4 m=-44$$

10. $$n^{2}-2 n=-3$$

11. $$r^{2}+6 r=-11$$

12. $$t^{2}-14 t=-50$$

13. $$a^{2}-10 a=-5$$

14. $$b^{2}+6 b=41$$

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

16. $$y^{2}-3 y=2$$

17. $$u^{2}-14 u+12=-1$$

18. $$z^{2}+2 z-5=2$$

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

20. $$t^{2}-10 t-6=5$$

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

22. $$w^{2}=5 w-1$$

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

24. $$y^{2}-14=6 y$$

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

26. $$(y+9)(y+7)=80$$

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

28. $$(x-2)(x-6)=5$$

Answer

5. $$u=-3, u=1$$

7. $$x=-1, x=21$$

9. $$m=-2 \pm 2 \sqrt{10} i$$

11. $$r=-3 \pm \sqrt{2} i$$

13. $$a=5 \pm 2 \sqrt{5}$$

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

17. $$u=1, u=13$$

19. $$r=-2, r=6$$

21. $$v=\frac{9}{2} \pm \frac{\sqrt{89}}{2}$$

23. $$x=5 \pm \sqrt{30}$$

25. $$x=-7, x=3$$

27. $$x=-5, x=-1$$

ExerciseS 29 - 40: Solve Quadratic Equations of the Form $$a x^{2}+b x+c=0$$ by Completing the Square

In the following exercises, solve by completing the square.

29. $$3 m^{2}+30 m-27=6$$

30. $$2 x^{2}-14 x+12=0$$

31. $$2 n^{2}+4 n=26$$

32. $$5 x^{2}+20 x=15$$

33. $$2 c^{2}+c=6$$

34. $$3 d^{2}-4 d=15$$

35. $$2 x^{2}+7 x-15=0$$

36. $$3 x^{2}-14 x+8=0$$

37. $$2 p^{2}+7 p=14$$

38. $$3 q^{2}-5 q=9$$

39. $$5 x^{2}-3 x=-10$$

40. $$7 x^{2}+4 x=-3$$

Answer

29. $$m=-11, m=1$$

31. $$n=1 \pm \sqrt{14}$$

33. $$c=-2, c=\frac{3}{2}$$

35. $$x=-5, x=\frac{3}{2}$$

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

39. $$x=\frac{3}{10} \pm \frac{\sqrt{191}}{10} i$$

ExerciseS 41 - 42: writing exercises

41. Solve the equation $$x^{2}+10 x=-25$$

1. by using the Square Root Property
2. by Completing the Square
3. Which method do you prefer? Why?

42. Solve the equation $$y^{2}+8y=48$$ by completing the square and explain all your steps.

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. After reviewing this checklist, what will you do to become confident for all objectives?

9.3E: 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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