18.4E: 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
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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. \(q=-\dfrac{3}{5}\)
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.
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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. \(2\)
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.
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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
ExerciseS 41 - 42: Writing Exercises
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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?