Start Up Day 38 12/15Solve Each Equation:

6443 x

OBJECTIVE:ESSENTIAL QUESTION: What is a Radical Equation? When you square each side of an equation, is the resulting equation equivalent to the original?

S.W.B.A.T. Solve equations containing radicals.

HOME LEARNING: P. 395 # 9, 12, 18, 22, 26, 29, 33, 38, 41, 49 & 52MathXLforSchool 40 Point Mid-Chapter Quiz (Covering Lessons 6.1 through 6.4). Due by 1/6

Section 6.5

Radical Equations & Extraneous Solutions

A radical equation is an equation in which the variable occurs in a square root, cube root, or any higher root. In this section, you will learn how to solve radical equations.

When the variable occurs in a square root, it is necessary to square both sides of the equation. When you square both sides of an equation, sometimes extra answers creep in, called extraneous roots. For example, consider the following very simple original equation.

24

22

xx

x Square both sides.Solve the equation. But -2 does notwork in the original equation. Then -2 is an extraneous root. It’s like a hitchhiker that we picked up when we squared both sides. It works only in the squared form and is not a root of the original.

Radical Equations

Just to note then….

When you solve a rational equation and must raise both sides to an even power, remember to check your roots. Throw out any extra (extraneous) roots from your solution set.

Another thing that you should know is … if your equation contains two or more square root expressions, you will need to isolate one square root expression, square both sides, and then you may have to repeat the process. You may have to square both sides twice.

Solving Radical Equations

Solving Radical Equations Containing nth Roots

1) If necessary, arrange terms so that one radical is isolated on one side of the equation.

2) Raise both sides of the equation to the nth power to eliminate the nth root.

3) Solve the resulting equation. If this equation still contains radicals, repeat steps 1 and 2.

4) Check all proposed solutions in the original equation.

Solving Radical EquationsEXAMPLE 1: EXAMPLE 1:

Solve: .843 x

SOLUTIONSOLUTION1) Isolate a radical on one side.

Simplify.

.843 x

2) Raise both sides to the nth power. Because n, the index, is 2, we square both sides.

22843 x

6443 x

Solving Radical Equations

This is the equation from step 2.

3) Solve the resulting equation.

6443 x

Subtract 4 from both sides.603 xDivide both sides by 3.20x

4) Check the proposed solution in the original equation.Check 20:

843 x

8 4203

8460

864

88 trueThe solution is 20.

The solution set is {20}.?

?

?

Solving Radical EquationsYOUR TURN: YOUR TURN:

Solve: .0523 x

SOLUTIONSOLUTION1) Isolate a radical on one side. The radical can be isolated by adding 5 to both sides. We obtain

Simplify.

.523 x

2) Raise both sides to the nth power.

2523 x

2523 x

Solving Radical Equations

3) Solve the resulting equation.

273 x9x

4) Check the proposed solution in the original equation.Check 9:

0523 x

0 5-293

0525

00 true

The solution is 9.

The solution set is {9}.?

?

2523 x

Solving Radical EquationsEXAMPLE 2: EXAMPLE 2:

Solve: .61152 x

SOLUTIONSOLUTION1) Isolate a radical on one side. The radical, , can be isolated by subtracting 11 from both sides. We obtain

Simplify.

52 x

.552 x

2) Raise both sides to the nth power. Because n, the index, is 2, we square both sides.

22552 x

2552 x

Solving Radical Equations

This is the equation from step 2.

3) Solve the resulting equation.2552 x

Subtract 5 from both sides.202 xDivide both sides by 2.10x

4) Check the proposed solution in the original equation.Check 10:

61152 x

6115102

61125

6115

616 falseTherefore there is no solution to the equation.

?

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?

Solving Radical EquationsEXAMPLE 3: EXAMPLE 3:

Solve: .5733 xx

SOLUTIONSOLUTION1) Isolate a radical on one side. The radical, , can be isolated by subtracting 3x from both sides. We obtain

Simplify. Use the special formula

73 x

.3573 xx

2) Raise both sides to the nth power. Because n, the index, is 2, we square both sides.

29302573 xxx

223573 xx

.2 222 BABABA

Solving Radical Equations

Equation from step 2.

3) Solve the resulting equation. Because of the -term, the resulting equation is a quadratic equation. We need to write this quadratic equation in standard form. We can obtain zero on the left side by subtracting 3x and 7 from both sides.

2530973 2 xxx

CONTINUECONTINUEDD 2x

Subtract 3x and 7 from both sides.

182790 2 xx

Factor out the GCF, 9. 2390 2 xx

Divide both sides by 9.230 2 xx

Factor the right side. 120 xx

Solving Radical EquationsCONTINUED: CONTINUED:

Set each factor equal to 0.02 x 01xSolve for x.2x 1x

4) Check the proposed solutions in the original equation.Check -2: Check -1:

5733 xx 5733 xx 572323 571313

516 543

516 523 57 55

The solution is -1. The solution set is {-1}.truefalse

?

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? ?

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Solving Radical EquationsYOUR TURN: YOUR TURN:

Solve: .712 xx

SOLUTIONSOLUTION

4) Check the proposed solution in the original equation.Check 12:

712 xx

7121122

525

55 true

?

?

12x

In Summary…Solving Radical Equations Containing nth Roots

1.Isolate one radical on one side of the equation.2.Raise both sides to the nth power3.Solve the resulting equation4.Check proposed solutions in the original equation.

Sometimes proposed solutions will work in the final simplified form of the original equation, but will not work in the original equation itself. These imposter roots that sometimes slip in when we square both sides of an equation are called extraneous roots.

Solving Radical Equations—Double Trouble!EXAMPLE 4: EXAMPLE 4:

Solve: .444 xx

SOLUTIONSOLUTION1) Isolate a radical on one side. The radical, , can be isolated by subtracting from both sides. We obtain

Simplify. Use the special formula

4x

2) Raise both sides to the nth power. Because n, the index, is 2, we square both sides.

448164 xxx .2 222 BABABA

4x.444 xx

22444 xx

Solving Radical Equations

Combine like terms.48204 xxxCONTINUECONTINUE

D D

1) Isolate a radical on one side. The radical, can be isolated by subtracting 20 + x from both sides and then dividing both sides by -8. We obtain

,4x

.43 x

2) Raise both sides to the nth power. Because n, the index, is 2, we square both sides.

22 43 x

49 x Square the 3 and the .4x

Solving Radical EquationsCONTINUECONTINUE

D D 3) Solve the resulting equation.

49 x This is the equation from the last step.

x5 Subtract 4 from both sides.

3) Check the proposed solution in the original equation.

444 xxCheck 5:

44545

491

431

44

The solution is 5. The solution set is {5}.

true

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Let’s try one more: p. 395 #18

Already Isolated…So Power UP! Why 3/2?

Simplify Each Side. Remember: We Multiply Exponents to raise a power to a power!

Simplify your radical expression.

Solve your equation.