# 10.2E: Exercises

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## 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 $$\PageIndex{43}$$

$$a^2+10a$$

$$(a+5)^2$$

##### Example $$\PageIndex{44}$$

$$b^2+12b$$

##### Example $$\PageIndex{45}$$

$$m^2+18m$$

$$(m+9)^2$$

##### Example $$\PageIndex{46}$$

$$n^2+16n$$

##### Example $$\PageIndex{47}$$

$$m^2−24m$$

$$(m−12)^2$$

##### Example $$\PageIndex{48}$$

$$n^2−16n$$​​​​​​​

##### Example $$\PageIndex{49}$$

$$p^2−22p$$

$$(p−11)^2$$

##### Example $$\PageIndex{50}$$

$$q^2−6q$$

##### Example $$\PageIndex{51}$$

$$x^2−9x$$

$$(x−\frac{9}{2})^2$$

##### Example $$\PageIndex{52}$$

$$y^2+11y$$

##### Example $$\PageIndex{53}$$

$$p^2−13p$$

$$(p−16)^2$$

##### 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 $$\PageIndex{55}$$

$$v^2+6v=40$$

$$v=−10$$, $$v=4$$

##### Example $$\PageIndex{56}$$

$$w^2+8w=65$$

##### Example $$\PageIndex{57}$$

$$u^2+2u=3$$

$$u=−3$$, $$u=1$$​​​​​​​

##### Example $$\PageIndex{58}$$

$$z^2+12z=−11$$

##### Example $$\PageIndex{59}$$

$$c^2−12c=13$$

$$c=−1$$, $$c=13$$

##### Example $$\PageIndex{60}$$

$$d^2−8d=9$$

##### Example $$\PageIndex{61}$$

$$x^2−20x=21$$

$$x=−1$$, $$x=21$$

##### Example $$\PageIndex{62}$$

$$y^2−2y=8$$

##### Example $$\PageIndex{63}$$

$$m^2+4m=−44$$

no real solution

##### Example $$\PageIndex{64}$$

$$n^2−2n=−3$$

##### Example $$\PageIndex{65}$$

$$r^2+6r=−11$$

no real solution

##### Example $$\PageIndex{66}$$

$$t^2−14t=−50$$

##### Example $$\PageIndex{67}$$

$$a^2−10a=−5$$

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

##### Example $$\PageIndex{68}$$

$$b^2+6b=41$$

##### Example $$\PageIndex{69}$$

$$u^2−14u+12=−1$$

$$u=1$$, $$u=13$$

##### Example $$\PageIndex{70}$$

$$z^2+2z−5=2$$

##### Example $$\PageIndex{71}$$

$$v^2=9v+2$$

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

##### Example $$\PageIndex{72}$$

$$w^2=5w−1$$

##### Example $$\PageIndex{73}$$

$$(x+6)(x−2)=9$$

$$x=−7$$, $$x=3$$

##### 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 $$\PageIndex{75}$$

$$3m^2+30m−27=6$$

$$m=−11$$, $$m=1$$

##### Example $$\PageIndex{76}$$

$$2n^2+4n−26=0$$

##### Example $$\PageIndex{77}$$

$$2c^2+c=6$$

$$c=−2$$, $$c=\frac{3}{2}$$

##### Example $$\PageIndex{78}$$

$$3d^2−4d=15$$

##### Example $$\PageIndex{79}$$

$$2p^2+7p=14$$

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

##### 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.

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?
1. −5
2. −5
​​​​​​​
##### 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?

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