9.6E: Exercises

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Practice Makes Perfect

Solve Radical Equations

In the following exercises, check whether the given values are solutions.

Example $\PageIndex{43}$

For the equation $\sqrt{x+12}=x$:

Is x=4 a solution?
Is x=−3 a solution?

Answer

Example $\PageIndex{44}$

For the equation $\sqrt{−y+20}=y$

Is y=4 a solution?
Is y=−5 a solution?

Example $\PageIndex{45}$

For the equation $\sqrt{t+6}=t$:

Is t=−2 a solution?
Is t=3 a solution?

Answer

Example $\PageIndex{46}$

For the equation $\sqrt{u+42}=u$:

Is u=−6 a solution?
Is u=7 a solution?

In the following exercises, solve.

Example $\PageIndex{47}$

$\sqrt{5y+1}=4$

Answer: 3

Example $\PageIndex{48}$

$\sqrt{7z+15}=6$

Example $\PageIndex{49}$

$\sqrt{5x−6}=8$

Answer: 14

Example $\PageIndex{50}$

$\sqrt{4x−3}=7$

Example $\PageIndex{51}$

$\sqrt{2m−3}−5=0$

Answer: 14

Example $\PageIndex{52}$

$\sqrt{2n−1}−3=0$

Example $\PageIndex{53}$

$\sqrt{6v−2}−10=0$

Answer: 17

Example $\PageIndex{54}$

$\sqrt{4u+2}−6=0$

Example $\PageIndex{55}$

$\sqrt{5q+3}−4=0$

Answer: $\frac{13}{5}$

Example $\PageIndex{56}$

$\sqrt{4m+2}+2=6$

Example $\PageIndex{57}$

$\sqrt{6n+1}+4=8$

Answer: $\frac{5}{2}$

Example $\PageIndex{58}$

$\sqrt{2u−3}+2=0$

Example $\PageIndex{59}$

$\sqrt{5v−2}+5=0$

Answer: no solution

Example $\PageIndex{60}$

$\sqrt{3z−5}+2=0$

Example $\PageIndex{61}$

$\sqrt{2m+1}+4=0$

Answer: no solution

Example $\PageIndex{62}$

$\sqrt{u−3}+3=u$
$\sqrt{x+1}−x+1=0$

Example $\PageIndex{63}$

$\sqrt{v−10}+10=v$
$\sqrt{y+4}−y+2=0$

Answer

10, 11
5

Example $\PageIndex{64}$

$\sqrt{r−1}−r=−1$
$\sqrt{z+100}−z+10=0$

Example $\PageIndex{65}$

$\sqrt{s−8}−s=−8$
$\sqrt{w+25}−w+5=0$

Answer

Example $\PageIndex{66}$

$3\sqrt{2x−3}−20=7$

Example $\PageIndex{67}$

$2\sqrt{5x+1}−8=0$

Answer: 3

Example $\PageIndex{68}$

$2\sqrt{8r+1}−8=2$

Example $\PageIndex{69}$

$3\sqrt{7y+1}−10=8$

Answer: 5

Example $\PageIndex{70}$

$\sqrt{3u−2}=\sqrt{5u+1}$

Example $\PageIndex{71}$

$\sqrt{4v+3}=\sqrt{v−6}$

Answer: not a real number

Example $\PageIndex{72}$

$\sqrt{8+2r}=\sqrt{3r+10}$

Example $\PageIndex{73}$

$\sqrt{12c+6}=\sqrt{10−4c}$

Answer: $\frac{1}{4}$

Example $\PageIndex{74}$

$\sqrt{a}+2=\sqrt{a+4}$
$\sqrt{b−2}+1=\sqrt{3b+2}$

Example $\PageIndex{75}$

$\sqrt{r}+6=\sqrt{r+8}$
$\sqrt{s−3}+2=\sqrt{s+4}$

Answer

no solution
$\frac{57}{16}$

Example $\PageIndex{76}$

$\sqrt{u}+1=\sqrt{u+4}$
$\sqrt{n−5}+4=\sqrt{3n+7}$

Example $\PageIndex{77}$

$\sqrt{x}+10=\sqrt{x+2}$
$\sqrt{y−2}+2=\sqrt{2y+4}$

Answer

no solution
6

Example $\PageIndex{78}$

$\sqrt{2y+4}+6=0$

Example $\PageIndex{79}$

$\sqrt{8u+1}+9=0$

Answer: no solution

Example $\PageIndex{80}$

$\sqrt{a}+1=\sqrt{a+5}$

Example $\PageIndex{81}$

$\sqrt{d}−2=\sqrt{d−20}$

Answer: 36

Example $\PageIndex{82}$

$\sqrt{6s+4}=\sqrt{8s−28}$

Example $\PageIndex{83}$

$\sqrt{9p+9}=\sqrt{10p−6}$

Answer: 15

Use Square Roots in Applications

In the following exercises, solve. Round approximations to one decimal place.

Example $\PageIndex{84}$

Landscaping Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of 75 square feet. Use the formula $s=\sqrt{A}$ to find the length of each side of his garden. Round your answer to the nearest tenth of a foot.

Example $\PageIndex{85}$

Landscaping Vince wants to make a square patio in his yard. He has enough concrete to pave an area of 130 square feet. Use the formula $s=\sqrt{A}$ to find the length of each side of his patio. Round your answer to the nearest tenth of a foot.

Answer: 11.4 feet

Example $\PageIndex{86}$

Gravity While putting up holiday decorations, Renee dropped a light bulb from the top of a 64 foot tall tree. Use the formula $t=\frac{\sqrt{h}}{4}$ to find how many seconds it took for the light bulb to reach the ground.

Example $\PageIndex{87}$

Gravity An airplane dropped a flare from a height of 1024 feet above a lake. Use the formula $t=\frac{\sqrt{h}}{4}$ to find how many seconds it took for the flare to reach the water.

Answer: 8 seconds

Example $\PageIndex{88}$

Gravity A hang glider dropped his cell phone from a height of 350 feet. Use the formula $t=\frac{\sqrt{h}}{4}$ to find how many seconds it took for the cell phone to reach the ground.

Example $\PageIndex{89}$

Gravity A construction worker dropped a hammer while building the Grand Canyon skywalk, 4000 feet above the Colorado River. Use the formula $t=\frac{\sqrt{h}}{4}$ to find how many seconds it took for the hammer to reach the river.

Answer: 15.8 seconds

Example $\PageIndex{90}$

Accident investigation The skid marks for a car involved in an accident measured 54 feet. Use the formula $s=\sqrt{24d}$ to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.

Example $\PageIndex{91}$

Accident investigation The skid marks for a car involved in an accident measured 216 feet. Use the formula $s=\sqrt{24d}$ to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.

Answer: 72 feet

Example $\PageIndex{92}$

Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 175 feet. Use the formula $s=\sqrt{24d}$ to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.

Example $\PageIndex{93}$

Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 117 feet. Use the formula $s=\sqrt{24d}$ to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.

Answer: 53.0 feet

Writing Exercises

Example $\PageIndex{94}$

Explain why an equation of the form $\sqrt{x}+1=0$ has no solution.

Example $\PageIndex{95}$

ⓐ Solve the equation $\sqrt{r+4}−r+2=0$.
ⓑ Explain why one of the “solutions” that was found was not actually a solution to the equation.

Answer: Answers will vary.

Self Check

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

$This table has two rows and four columns. The first row labels each column, “I can…,” “Confidently,” “With some help,” and “No minus I don’t get it!” The row under “I can…,” reads, “use square roots in applications.” All the other rows are empty.$

ⓑ 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 \(\PageIndex{49}\)

Example \(\PageIndex{50}\)

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Example \(\PageIndex{52}\)

Example \(\PageIndex{53}\)

Example \(\PageIndex{54}\)

Example \(\PageIndex{55}\)

Example \(\PageIndex{56}\)

Example \(\PageIndex{57}\)

Example \(\PageIndex{58}\)

Example \(\PageIndex{59}\)

Example \(\PageIndex{60}\)

Example \(\PageIndex{61}\)

Example \(\PageIndex{62}\)

Example \(\PageIndex{63}\)

Example \(\PageIndex{64}\)

Example \(\PageIndex{65}\)

Example \(\PageIndex{66}\)

Example \(\PageIndex{67}\)

Example \(\PageIndex{68}\)

Example \(\PageIndex{69}\)

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 \(\PageIndex{80}\)

Example \(\PageIndex{81}\)

Example \(\PageIndex{82}\)

Example \(\PageIndex{83}\)

Example \(\PageIndex{84}\)

Example \(\PageIndex{85}\)

Example \(\PageIndex{86}\)

Example \(\PageIndex{87}\)

Example \(\PageIndex{88}\)

Example \(\PageIndex{89}\)

Example \(\PageIndex{90}\)

Example \(\PageIndex{91}\)

Example \(\PageIndex{92}\)

Example \(\PageIndex{93}\)

Writing Exercises

Example \(\PageIndex{94}\)

Example \(\PageIndex{95}\)

Self Check