5.4E: Exercises

Practice Makes Perfect

Exercise $\PageIndex{11}$ Solve equations in quadratic form

In the following exercises, solve.

1. $x^{4}-7 x^{2}+12=0$
2. $x^{4}-9 x^{2}+18=0$
3. $x^{4}-13 x^{2}-30=0$
4. $x^{4}+5 x^{2}-36=0$
5. $2 x^{4}-5 x^{2}+3=0$
6. $4 x^{4}-5 x^{2}+1=0$
7. $2 x^{4}-7 x^{2}+3=0$
8. $3 x^{4}-14 x^{2}+8=0$
9. $(x-3)^{2}-5(x-3)-36=0$
10. $(x+2)^{2}-3(x+2)-54=0$
11. $(3 y+2)^{2}+(3 y+2)-6=0$
12. $(5 y-1)^{2}+3(5 y-1)-28=0$
13. $\left(x^{2}+1\right)^{2}-5\left(x^{2}+1\right)+4=0$
14. $\left(x^{2}-4\right)^{2}-4\left(x^{2}-4\right)+3=0$
15. $2\left(x^{2}-5\right)^{2}-5\left(x^{2}-5\right)+2=0$
16. $2\left(x^{2}-5\right)^{2}-7\left(x^{2}-5\right)+6=0$
17. $x-\sqrt{x}-20=0$
18. $x-8 \sqrt{x}+15=0$
19. $x+6 \sqrt{x}-16=0$
20. $x+4 \sqrt{x}-21=0$
21. $6 x+\sqrt{x}-2=0$
22. $6 x+\sqrt{x}-1=0$
23. $10 x-17 \sqrt{x}+3=0$
24. $12 x+5 \sqrt{x}-3=0$
25. $x^{\frac{2}{3}}+9 x^{\frac{1}{3}}+8=0$
26. $x^{\frac{2}{3}}-3 x^{\frac{1}{3}}=28$
27. $x^{\frac{2}{3}}+4 x^{\frac{1}{3}}=12$
28. $x^{\frac{2}{3}}-11 x^{\frac{1}{3}}+30=0$
29. $6 x^{\frac{2}{3}}-x^{\frac{1}{3}}=12$
30. $3 x^{\frac{2}{3}}-10 x^{\frac{1}{3}}=8$
31. $8 x^{\frac{2}{3}}-43 x^{\frac{1}{3}}+15=0$
32. $20 x^{\frac{2}{3}}-23 x^{\frac{1}{3}}+6=0$
33. $x-8 x^{\frac{1}{2}}+7=0$
34. $2 x-7 x^{\frac{1}{2}}=15$
35. $6 x^{-2}+13 x^{-1}+5=0$
36. $15 x^{-2}-26 x^{-1}+8=0$
37. $8 x^{-2}-2 x^{-1}-3=0$
38. $15 x^{-2}-4 x^{-1}-4=0$

Answer

1. $x=\pm \sqrt{3}, x=\pm 2$

3. $x=\pm \sqrt{15}, x=\pm \sqrt{2} i$

5. $x=\pm 1, x=\frac{ \pm \sqrt{6}}{2}$

7. $x=\pm \sqrt{3}, x=\pm \frac{\sqrt{2}}{2}$

9. $x=-1, x=12$

11. $x=-\frac{5}{3}, x=0$

13. $x=0, x=\pm \sqrt{3}$

15. $x=\pm \frac{11}{2}, x=\pm \frac{\sqrt{22}}{2}$

17. $x=25$

19. $x=4$

21. $x=\frac{1}{4}$

23. $x=\frac{1}{25}, x=\frac{9}{4}$

25. $x=-1, x=-512$

27. $x=8, x=-216$

29. $x=\frac{27}{8}, x=-\frac{64}{27}$

31. $x=27, x=64,000$

33. $x=1, x=49$

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

37. $x=-2, x=\frac{4}{3}$

Exercise $\PageIndex{12}$ writing exercises

1. Explain how to recognize an equation in quadratic form.
2. Explain the procedure for solving an equation in quadratic form.

Answer

1. Answers will vary.

Self Check

a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

Figure 9.4.43

b. On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?