
9.3E: Exercises


Practice Makes Perfect

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

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

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

27. $$b=\dfrac{-2 \pm \sqrt{11}}{6}$$

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

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

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. $$7 y^{2}+2 y+8=0$$

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

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

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. $$\left(w+\dfrac{4}{3}\right)^{2}=\dfrac{2}{9}$$

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

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

1. Solve the equation $$x^{2}+10 x=120$$
1. by completing the square
3. Which method do you prefer? Why?
2. Solve the equation $$12 y^{2}+23 y=24$$
1. by completing the square