9.4E: 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. \(\dfrac{1}{4} m^{2}+\dfrac{1}{12} m=\dfrac{1}{3}\)
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
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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. \(q=-\dfrac{3}{5}\)
In the following exercises, determine the number of real solutions for each quadratic equation.
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- \(4 x^{2}-5 x+16=0\)
- \(36 y^{2}+36 y+9=0\)
- \(6 m^{2}+3 m-5=0\)
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- \(9 v^{2}-15 v+25=0\)
- \(100 w^{2}+60 w+9=0\)
- \(5 c^{2}+7 c-10=0\)
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- \(r^{2}+12 r+36=0\)
- \(8 t^{2}-11 t+5=0\)
- \(3 v^{2}-5 v-1=0\)
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- \(25 p^{2}+10 p+1=0\)
- \(7 q^{2}-3 q-6=0\)
- \(7 y^{2}+2 y+8=0\)
- Answer
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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.
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- \(x^{2}-5 x-24=0\)
- \((y+5)^{2}=12\)
- \(14 m^{2}+3 m=11\)
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- \((8 v+3)^{2}=81\)
- \(w^{2}-9 w-22=0\)
- \(4 n^{2}-10=6\)
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- \(6 a^{2}+14=20\)
- \(\left(x-\dfrac{1}{4}\right)^{2}=\dfrac{5}{16}\)
- \(y^{2}-2 y=8\)
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- \(8 b^{2}+15 b=4\)
- \(\dfrac{5}{9} v^{2}-\dfrac{2}{3} v=1\)
- \(\left(w+\dfrac{4}{3}\right)^{2}=\dfrac{2}{9}\)
- Answer
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37. a. Factor b. Square Root c. Quadratic Formula
39. a. Quadratic Formula b. Square Root c. Factor
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Solve the equation \(x^{2}+10 x=120\)
- by completing the square
- using the Quadratic Formula
- Which method do you prefer? Why?
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Solve the equation \(12 y^{2}+23 y=24\)
- by completing the square
- using the Quadratic Formula
- Which method do you prefer? Why?
- Answer
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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?