13.8.6E: Exercises
-
- Last updated
- Save as PDF
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
-
- yes
- no
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
-
- no
- yes
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
-
- 8,9
- 11
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
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.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?