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 $\PageIndex{43}$

$a^2+10a$

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 $x^2+10x=−2$

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

Answer

−5
−5
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.

$This table has four rows and four columns. The first row is a header row and it labels each column. The first column is labeled “I can ...”, the second “Confidently”, the third “With some help” and the last “No–I don’t get it”. In the “I can...” column the next row reads “complete the square of a binomial expression.” The next row reads “solve quadratic equations of the form x squared plus b x plus c equals zero by completing the square.” and the last row reads “solve quadratic equations of the form a x squared plus b x plus c equals zero by completing the square.” The remaining columns are blank.$

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

Practice Makes Perfect

Example \(\PageIndex{43}\)

Example \(\PageIndex{44}\)

Example \(\PageIndex{45}\)

Example \(\PageIndex{46}\)

Example \(\PageIndex{47}\)

Example \(\PageIndex{48}\)

Example \(\PageIndex{49}\)

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Example \(\PageIndex{52}\)

Example \(\PageIndex{53}\)

Example \(\PageIndex{54}\)

Example \(\PageIndex{55}\)

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Example \(\PageIndex{60}\)

Example \(\PageIndex{61}\)

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Example \(\PageIndex{63}\)

Example \(\PageIndex{64}\)

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Example \(\PageIndex{66}\)

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Example \(\PageIndex{70}\)

Example \(\PageIndex{71}\)

Example \(\PageIndex{72}\)

Example \(\PageIndex{73}\)

Example \(\PageIndex{74}\)

Example \(\PageIndex{75}\)

Example \(\PageIndex{76}\)

Example \(\PageIndex{77}\)

Example \(\PageIndex{78}\)

Example \(\PageIndex{79}\)

Example \(\PageIndex{80}\)

Everyday Math

Example \(\PageIndex{81}\)

Example \(\PageIndex{82}\)

Writing Exercises

Example \(\PageIndex{83}\)

Example \(\PageIndex{84}\)

Self Check