18.3E: 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.
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- \(m^{2}-24 m\)
- \(x^{2}-11 x\)
- \(p^{2}-\frac{1}{3} p\)
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- \(n^{2}-16 n\)
- \(y^{2}+15 y\)
- \(q^{2}+\frac{3}{4} q\)
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- \(p^{2}-22 p\)
- \(y^{2}+5 y\)
- \(m^{2}+\frac{2}{5} m\)
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- \(q^{2}-6 q\)
- \(x^{2}-7 x\)
- \(n^{2}-\frac{2}{3} n\)
- Answer
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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
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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
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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 \(x^{2}+10 x=-25\)
- by using the Square Root Property
- by Completing the Square
- Which method do you prefer? Why?
42. Solve the equation \(y^{2}+8y=48\) by completing the square and explain all your steps.
- 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. After reviewing this checklist, what will you do to become confident for all objectives?