1. Chapter 8 Section 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. Solving Equations with Radicals Solve radical equations having square root radicals. Identify equations with no solutions. Solve equations by squaring a binomial. Solve radical equations having cube root radicals. 1 1 4 4 3 3 2 28.68.6

3. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Equations with Radicals. A radical equation is an equation having a variable in the radicand, such as Slide 8.6 - 3 or

4. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 Objective 1 Slide 8.6 - 4 Solve radical equations having square root radicals.

5. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To solve radical equations having square root radicals, we need a new property, called the squaring property of equality. Be very careful with the squaring property: Using this property can give a new equation with more solutions than the original equation has. Because of this possibility, checking is an essential part of the process. All proposed solutions from the squared equation must be checked in the original equation. Slide 8.6 - 5 Solve radical equations having square root radicals. If each side of a given equation is squared, then all solutions of the original equation are among the solutions of the squared equation.

6. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Solve. Solution: Using the Squaring Property of Equality Slide 8.6 - 6 It is important to note that even though the algebraic work may be done perfectly, the answer produced may not make the original equation true.

7. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve. EXAMPLE 2 Using the Squaring Property with a Radical on Each Side Slide 8.6 - 7 Solution:

8. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Objective 2 Identify equations with no solutions. Slide 8.6 - 8

9. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Solution: Using the Squaring Property when One Side Is Negative Slide 8.6 - 9 Solve. False Because represents the principal or nonnegative square root of x in Example 3, we might have seen immediately that there is no solution. Check:

10. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Use the following steps when solving an equation with radicals. Step 1Isolate a radical. Arrange the terms so that a radical is isolated on one side of the equation. Solving a Radical Equation. Slide 8.6 - 10 Step 6Check all proposed solutions in the original equation. Step 5Solve the equation. Find all proposed solutions. Step 4Repeat Steps 1-3 if there is still a term with a radical. Step 3Combine like terms. Step 2Square both sides.

11. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solution: Using the Squaring Property with a Quadratic Expression Slide 8.6 - 11 Solve Since x must be a positive number the solution set is Ø.

12. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3 Objective 3 Slide 8.6 - 12 Solve equations by squaring a binomial.

13. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solve Solution: Using the Squaring Property when One Side Has Two Terms Slide 8.6 - 13 Since x must be positive the solution set is {4}. or

14. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve. EXAMPLE 6 Rewriting an Equation before using the Squaring Property Slide 8.6 - 14 Solution: The solution set is {4,9}. or

15. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve equations by squaring a binomial. Errors often occur when both sides of an equation are squared. For instance, when both sides of are squared, the entire binomial 2 x + 1 must be squared to get 4 x 2 + 4 x + 1. It is incorrect to square the 2 x and the 1 separately to get 4 x 2 + 1. Slide 8.6 - 15

16. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Using the Squaring Property Twice Slide 8.6 - 16 Solve. Solution: The solution set is {8}.

17. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4 Objective 4 Slide 8.6 - 17 Solve radical equations having cube root radicals.

18. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve radical equations having cube root radicals. Slide 8.6 - 18 We can extend the concept of raising both sides of an equation to a power in order to solve radical equations with cube roots.

19. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Solving Equations with Cube Root Radicals Slide 8.6 - 19 Solve each equation. Solution: or