9.2E: Exercises

Practice Makes Perfect

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.

Example $\PageIndex{43}$

$a^2+10a$

Answer

$(a+5)^2$

Example $\PageIndex{44}$

$b^2+12b$

Example $\PageIndex{45}$

$m^2+18m$

Answer

$(m+9)^2$

Example $\PageIndex{46}$

$n^2+16n$

Example $\PageIndex{47}$

$m^2−24m$

Answer

$(m−12)^2$

Example $\PageIndex{48}$

$n^2−16n$​​​​​​​

Example $\PageIndex{49}$

$p^2−22p$

Answer

$(p−11)^2$

Example $\PageIndex{50}$

$q^2−6q$

Example $\PageIndex{51}$

$x^2−9x$

Answer

$(x−\frac{9}{2})^2$

Example $\PageIndex{52}$

$y^2+11y$

Example $\PageIndex{53}$

$p^2−13p$

Answer

$(p−16)^2$

Example $\PageIndex{54}$

$q^2+34q$

​​​​​​​Solve Quadratic Equations of the Form \(x^2+bx+c=0\) by Completing the Square

In the following exercises, solve by completing the square.

Example $\PageIndex{55}$

$v^2+6v=40$

Answer

$v=−10$, $v=4$

Example $\PageIndex{56}$

$w^2+8w=65$

Example $\PageIndex{57}$

$u^2+2u=3$

Answer

$u=−3$, $u=1$​​​​​​​

Example $\PageIndex{58}$

$z^2+12z=−11$

Example $\PageIndex{59}$

$c^2−12c=13$

Answer

$c=−1$, $c=13$

Example $\PageIndex{60}$

$d^2−8d=9$

Example $\PageIndex{61}$

$x^2−20x=21$

Answer

$x=−1$, $x=21$

Example $\PageIndex{62}$

$y^2−2y=8$

Example $\PageIndex{63}$

$m^2+4m=−44$

Answer

no real solution

Example $\PageIndex{64}$

$n^2−2n=−3$

Example $\PageIndex{65}$

$r^2+6r=−11$

Answer

no real solution

Example $\PageIndex{66}$

$t^2−14t=−50$

Example $\PageIndex{67}$

$a^2−10a=−5$

Answer

$a=5\pm2\sqrt{5}$

Example $\PageIndex{68}$

$b^2+6b=41$

Example $\PageIndex{69}$

$u^2−14u+12=−1$

Answer

$u=1$, $u=13$

Example $\PageIndex{70}$

$z^2+2z−5=2$

Example $\PageIndex{71}$

$v^2=9v+2$

Answer

$v=\frac{9}{2}\pm\frac{\sqrt{89}}{2}$

Example $\PageIndex{72}$

$w^2=5w−1$

Example $\PageIndex{73}$

$(x+6)(x−2)=9$

Answer

$x=−7$, $x=3$

Example $\PageIndex{74}$

$(y+9)(y+7)=79$

Solve Quadratic Equations of the Form \(ax^2+bx+c=0\) by Completing the Square

In the following exercises, solve by completing the square.

Example $\PageIndex{75}$

$3m^2+30m−27=6$

Answer

$m=−11$, $m=1$

Example $\PageIndex{76}$

$2n^2+4n−26=0$

Example $\PageIndex{77}$

$2c^2+c=6$

Answer

$c=−2$, $c=\frac{3}{2}$

Example $\PageIndex{78}$

$3d^2−4d=15$

Example $\PageIndex{79}$

$2p^2+7p=14$

Answer

$p=−\frac{7}{4}\pm\frac{\sqrt{161}}{4}$

Example $\PageIndex{80}$

$3q^2−5q=9$

Everyday Math

Example $\PageIndex{81}$

Rafi is designing a rectangular playground to have an area of 320 square feet. He wants one side of the playground to be four feet longer than the other side. Solve the equation $p^2+4p=320$ for p, the length of one side of the playground. What is the length of the other side.

Answer

16 feet, 20 feet

Example $\PageIndex{82}$

Yvette wants to put a square swimming pool in the corner of her backyard. She will have a 3 foot deck on the south side of the pool and a 9 foot deck on the west side of the pool. She has a total area of 1080 square feet for the pool and two decks. Solve the equation $(s+3)(s+9)=1080$ for s, the length of a side of the pool.

Writing Exercises

Example $\PageIndex{83}$

Solve the equation $x^2+10x=−2$

1. by using the Square Root Property and
2. by completing the square.
3. Which method do you prefer? Why?

Answer

1. −5
2. −5
3. Answers will vary.

​​​​​​​

Example $\PageIndex{84}$

Solve the equation $y^2+8y=48$ by completing the square and explain all your steps.

​​​​​​​

Self Check

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

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