18.3E: Exercises

Last updated

Aug 13, 2020
Save as PDF
- 18.3: Solve Quadratic Equations by Completing the Square
- 18.4: Solve Quadratic Equations Using the Quadratic Formula

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\id}{\mathrm{id}}$ $\newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$ $\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$ $\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\id}{\mathrm{id}}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\kernel}{\mathrm{null}\,}$

$\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$

$\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$

$\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$ $\newcommand{\AA}{\unicode[.8,0]{x212B}}$

$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vectorC}[1]{\textbf{#1}}$

$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$

$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$

$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\avec}{\mathbf a}$

$\newcommand{\bvec}{\mathbf b}$

$\newcommand{\cvec}{\mathbf c}$

$\newcommand{\dvec}{\mathbf d}$

$\newcommand{\dtil}{\widetilde{\mathbf d}}$

$\newcommand{\evec}{\mathbf e}$

$\newcommand{\fvec}{\mathbf f}$

$\newcommand{\nvec}{\mathbf n}$

$\newcommand{\pvec}{\mathbf p}$

$\newcommand{\qvec}{\mathbf q}$

$\newcommand{\svec}{\mathbf s}$

$\newcommand{\tvec}{\mathbf t}$

$\newcommand{\uvec}{\mathbf u}$

$\newcommand{\vvec}{\mathbf v}$

$\newcommand{\wvec}{\mathbf w}$

$\newcommand{\xvec}{\mathbf x}$

$\newcommand{\yvec}{\mathbf y}$

$\newcommand{\zvec}{\mathbf z}$

$\newcommand{\rvec}{\mathbf r}$

$\newcommand{\mvec}{\mathbf m}$

$\newcommand{\zerovec}{\mathbf 0}$

$\newcommand{\onevec}{\mathbf 1}$

$\newcommand{\real}{\mathbb R}$

$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$

$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$

$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$

$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$

$\newcommand{\bcal}{\cal B}$

$\newcommand{\ccal}{\cal C}$

$\newcommand{\scal}{\cal S}$

$\newcommand{\wcal}{\cal W}$

$\newcommand{\ecal}{\cal E}$

$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$

$\newcommand{\gray}[1]{\color{gray}{#1}}$

$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$

$\newcommand{\rank}{\operatorname{rank}}$

$\newcommand{\row}{\text{Row}}$

$\newcommand{\col}{\text{Col}}$

$\renewcommand{\row}{\text{Row}}$

$\newcommand{\nul}{\text{Nul}}$

$\newcommand{\var}{\text{Var}}$

$\newcommand{\corr}{\text{corr}}$

$\newcommand{\len}[1]{\left|#1\right|}$

$\newcommand{\bbar}{\overline{\bvec}}$

$\newcommand{\bhat}{\widehat{\bvec}}$

$\newcommand{\bperp}{\bvec^\perp}$

$\newcommand{\xhat}{\widehat{\xvec}}$

$\newcommand{\vhat}{\widehat{\vvec}}$

$\newcommand{\uhat}{\widehat{\uvec}}$

$\newcommand{\what}{\widehat{\wvec}}$

$\newcommand{\Sighat}{\widehat{\Sigma}}$

$\newcommand{\lt}{<}$

$\newcommand{\gt}{>}$

$\newcommand{\amp}{&}$

$\definecolor{fillinmathshade}{gray}{0.9}$

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.

1. $m^{2}-24 m$
2. $x^{2}-11 x$
3. $p^{2}-\frac{1}{3} p$
1. $n^{2}-16 n$
2. $y^{2}+15 y$
3. $q^{2}+\frac{3}{4} q$
1. $p^{2}-22 p$
2. $y^{2}+5 y$
3. $m^{2}+\frac{2}{5} m$
1. $q^{2}-6 q$
2. $x^{2}-7 x$
3. $n^{2}-\frac{2}{3} n$

Answer

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

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

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: 41. Answers will vary

Self Check

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

This table provides a checklist to evaluate mastery of the objectives of this section. Choose how would you respond to the statement â€œI can complete the square of a binomial expression.â€ â€œConfidently,â€ â€œwith some help,â€ or â€œNo, I donâ€™t get it.â€ Choose how would you respond to the statement â€œI can solve quadratic equations of the form x squared plus b times x plus c equals 0 by completing the square.â€ â€œConfidently,â€ â€œwith some help,â€ or â€œNo, I donâ€™t get it.â€ Choose how would you respond to the statement â€œI can solve quadratic equations of the form a times x squared plus b times x plus c equals 0 by completing the square.â€ â€œConfidently,â€ â€œwith some help,â€ or â€œNo, I donâ€™t get it.â€ — Figure 9.2.103

b. After reviewing this checklist, what will you do to become confident for all objectives?

Search

Text Color

Text Size

Margin Size

Font Type

Practice Makes Perfect

ExerciseS 1 - 4: Complete the Square of a Binomial Expression

ExerciseS 5 - 28: Solve Quadratic Equations of the Form $x^{2}+b x+c=0$ by Completing the Square

ExerciseS 29 - 40: Solve Quadratic Equations of the Form $a x^{2}+b x+c=0$ by Completing the Square

ExerciseS 41 - 42: writing exercises

Self Check

Practice Makes Perfect

ExerciseS 1 - 4: Complete the Square of a Binomial Expression

ExerciseS 5 - 28: Solve Quadratic Equations of the Form x2+bx+c=0x^{2}+b x+c=0 by Completing the Square

ExerciseS 29 - 40: Solve Quadratic Equations of the Form ax2+bx+c=0a x^{2}+b x+c=0 by Completing the Square

ExerciseS 41 - 42: writing exercises

Self Check

Support Center

How can we help?

ExerciseS 5 - 28: Solve Quadratic Equations of the Form $x^{2}+b x+c=0$ by Completing the Square

ExerciseS 29 - 40: Solve Quadratic Equations of the Form $a x^{2}+b x+c=0$ by Completing the Square