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

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1 Copyright 008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 8 Section 6

2 8.6 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. Copyright 008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

3 Solving Equations with Radicals. A radical equation is an equation having a variable in the radicand, such as 1 3 or Copyright 008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8.6-3

4 Objective 1 Solve radical equations having square root radicals. Copyright 008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8.6-4

5 Solve radical equations having square root radicals. To solve radical equations having square root radicals, we need a new property, called the squaring property of equality. If each side of a given equation is squared, then all solutions of the original equation are among the solutions of the squared equation. 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. Copyright 008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8.6-5

6 EXAMPLE 1 Using the Squaring Property of Equality Solve. Solution: 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. Copyright 008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8.6-6

7 EXAMPLE Using the Squaring Property with a Radical on Each Side Solve. 3 9 Solution: Copyright 008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8.6-7

8 Objective Identify equations with no solutions. Copyright 008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8.6-8

9 EXAMPLE 3 Using the Squaring Property when One Side Is Negative Solve. Solution: 4 4 Check: False Because represents the principal or nonnegative square root of in Eample 3, we might have seen immediately that there is no solution. Copyright 008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8.6-9

10 Solving a Radical Equation. Use the following steps when solving an equation with radicals. Step 1 Step Step 3 Step 4 Step 5 Step 6 Isolate a radical. Arrange the terms so that a radical is isolated on one side of the equation. Square both sides. Combine like terms. Repeat Steps 1-3 if there is still a term with a radical. Solve the equation. Find all proposed solutions. Check all proposed solutions in the original equation. Copyright 008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide

11 EXAMPLE 4 Using the Squaring Property with a Quadratic Epression Solve Solution: Since must be a positive number the solution set is Ø. Copyright 008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide

12 Objective 3 Solve equations by squaring a binomial. Copyright 008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8.6-1

13 EXAMPLE 5 Using the Squaring Property when One Side Has Two Terms Solve Solution: or Since must be positive the solution set is {4}. Copyright 008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide

14 EXAMPLE 6 Rewriting an Equation before using the Squaring Property Solve. 5 6 Solution: or Copyright 008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The solution set is {4,9}. Slide

15 Solve equations by squaring a binomial. Errors often occur when both sides of an equation are squared. For instance, when both sides of 9 1 are squared, the entire binomial + 1 must be squared to get It is incorrect to square the and the 1 separately to get Copyright 008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide

16 EXAMPLE 7 Using the Squaring Property Twice Solve. Solution: The solution set is {8} Copyright 008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide

17 Objective 4 Solve radical equations having cube root radicals. Copyright 008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide

18 Solve radical equations having cube root radicals. We can etend the concept of raising both sides of an equation to a power in order to solve radical equations with cube roots. Copyright 008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide

19 EXAMPLE 8 Solving Equations with Cube Root Radicals Solve each equation. Solution: or ,1 Copyright 008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide