HFCC Math Lab Intermediate Algebra 18 SOLVING RADICAL EQUATIONS

Transcription

1 HFCC Math Lab Intermediate Algebra 18 SOLVING RADICAL EQUATIONS You already know how to solve linear equations and quadratic equations by factoring. In this handout, you are going to learn how to solve radical equations (1) containing one radical epression and () containing two radical epressions. But first let us answer the question: What is a radical equation? The answer is: a radical equation is an equation in which the variable we are solving for appears inside the radical symbol. For eample, 1, 5y 5, and 7 5 are all radical equations. Let us now proceed to solve radical equations. Part 1 Radical Equations Containing One Radical Epression Follow the following steps to solve such radical equations. Step I. Isolate the radical epression by itself on one side of the equation. Step II. Get rid of the radical by raising both sides to the power where n denotes the inde of the radical. In simple English, it means that we square both sides if the radical is the principal square root, we cube both sides if the radical is the cube root, we raise both sides to the power if the radical is the principal fourth root, and so on. Step III. Step IV. Step V. Simplify both sides of the equation in Step II and solve the resulting equation. Check the values obtained in Step III if the inde of the radical in the original equation is even. There is no need to check the values obtained in Step III if the inde of the radical in the original equation is odd. The solutions to the original equation are those values of the variable from Step IV which checked into the original equation. Revised 11/09 1

2 Please read the following eamples very carefully. Eamples: Solve each of the following radical equations 1. 5 Solution: To isolate the radical, we add 5 to both sides = Square both sides. = 9 Solve this linear equation for. Subtract from both sides So, Check 5 Replace by - ( ) Therefore, the solution is = Revised 11/09

3 . 5 Proceed eactly like the first eample. Add to both sides. 5 Since the principal square root (in fact, the principal even root) of any epression cannot be negative, the original equation has no solution. NOTE: If you had continued after isolating the radical, you will get =. But then = will not satisfy the original equation. So, once again, the original equation has no solution. NOTE:. Do you understand everything written in this handout so far? If your answer is no, please read again and then proceed further. Solution: To isolate the radical, we subtract from both sides. = = 5 Cube both sides Subtract from both sides NOTE: Since the equation involved the cube root, there is no need to check the solution.. 5 Subtract from both sides. Revised 11/09

4 5 Square both sides This is a quadratic equation. Interchange the two sides Get zero on the right side Factor the left side So, 0 or 7 0 or 7 Check for = Check for = Replace by. We get Replace by 7. We get false 7 7 true Since = does not check and = 7 checks, therefore the only solution to the original equation is = 7. Note: = called an etraneous solution Revised 11/09

5 Part II Radical equations containing two radical epressions when both the radicals are principal square roots. Follow the following steps to solve such radical equations. STEP I STEP II STEP III STEP IV STEP V Write the equation so that one radical epression (involving the principal square root) appears on each side of the equation. Square both sides of the equation very carefully. Solve the equation obtained in Step II. If one radical epression (involving the principal square root) still remains, solve this equation as we did in Part I of this handout. Check the values obtained in Step III in the original equation. The solutions to the original equation are those values of the variable from Step IV which checked into the original equation. Please read the following eamples very carefully. Eamples: Solve each of the following radical equations. 1. p 5p 0 Solution: First move one of the two radicals to the right side of the equation. So, we add 5p to both sides and obtain p 5p Square both sides p 5p p 5p Solve this linear equation p 5 p 8p Check: p 5p 0 p Replace p by we get Revised 11/09 5

6 Since the principal square root of a negative number is not a real number, therefore the equation has no solution.. p 5p 0 Proceed just like the first eample and get p. Check: p 5p 0 Replace p by we get true The solution to the equation is p. 1 First move one of the two radicals to the right side of the equation. So, we add to both sides and get 1 Revised 11/09 6

7 Square both sides 1 Be careful when you square on the right side. 1 1 Use FOIL on the right side. 1 5 We are now left with one radical epression. Isolate this radical epression on one side. Subtract and subtract 5 from both sides. 5 8 Divide both sides by -. 8 Square both sides to get 16 So, 1 Check: The solution is = 1 1 1( true) Revised 11/09 7

8 EXERCISES Solve each of the following radical equations y 5y 10. p 6 p y 5 5 y 19. n n 0. y y 7 1. p 1 p 9. y y y y Revised 11/09 8

9 ANSWERS AND SOLUTIONS 1. 7 Add 7 to both sides. 5 Square both sides to get =5 8 7 Check: true The solution is Add to both sides. 5 Since the principal square root of a number cannot be negative, the equation has no solution.. No solution. Revised 11/09 9

10 5. 5 Subtract from both sides. Square both sides. We get This is a quadratic equation Factor to get 0 0 or 0 or Check for Check for , Subtract from both sides. 1 Cube both sides (True) 5=5(True) The two solutions are =, The solution is 7 Revised 11/09 10

11 y 5y Square both sides. y y 5y 0 5y y 1 y 0 y 1 0or y 0 y 1or y Show yourself that both these values check. Therefore,the two solutions are y 1, y 10. p is the only solution ( p does not check) Subtract 7 from both sides Square both sides or 10 0 or 10 Show yourself that = does not check and = 10 checks. So, the only solution to the equation is = is the only solution does not check Revised 11/09 11

12 Subtract from both sides Proceed eactly like problem 11. You will get or 10 Show yourself that checks and 10 does not check. So, the only solution to the equation is is the only solution 1does not check Subtract 6 from both sides. 8 1 Raise both sides to the power. We get or 0 or Show yourself that both these values check. So, the two solutions are, 16. No solution. Revised 11/09 1

13 Add to both sides Square both sides Divide both sides by Square both sides This is a quadratic equation 0 Factor the left side 0 0or 0 0 or Show yourself that both these values check. Thus, the two solutions are 0, 18. y Revised 11/09 1

14 19. n n Add n to both sides. n+ n Square both sides. n n n n n n n n n n n n Subtract n and subtract from both sides. n n n n 1 n Square both sides n 1 n n n n 1 1 n n n 1 16 n n n 1 16 This is a quadratic equation. Write it in standard form. n n n n 1n 0 n n 11 0 n 0 or n 11 0 n or n 11 Show yourself that both these values check. Thus, the two solutions are n, n y Revised 11/09 1

15 1. p 1 p 9 Square both sides. p 1 p 9 p 1 p 1 p 9 p 1 p 1 p 1 p 9 5 p p 1 p 9 Subtract 5 and subtract p from both sides. We get p 1 p 9 5 p p 1 p Square both sides p 1 p p p p 16 p 1 p 8p 1 16 p 16 p 8 p 16 p p p p 8p 0 p p 8 0 So, p 0or p 8 0 Show yourself that both these values check. 6 Therefore, the two solutions are p 0, p 8. y Revised 11/09 15

16 . 1 y y 5 0 Add y+5 to both sides 1-y y 5 Square both sides 1 y y 5 1 y y y y y So, y Show that y checks The solution is y 9., NOTE: You can get additional instruction and practice by going to the following web sites: This website gives an ecellent detailed eample on solving radical equations. This website has some very good eamples on solving radical equations with two radicals with some graphical illustration. Revised 11/09 16