# Chapter 8 Section 6 8 6 Solving Equations

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Chapter 8 Section 6

8. 6 Solving Equations with Radicals Objectives 1 • Solve radical equations having square root radicals. 2 • Identify equations with no solutions. 3 • Solve equations by squaring a binomial. 4 • Solve radical equations having cube root radicals. Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Solving Equations with Radicals. A radical equation is an equation having a variable in the radicand, such as or Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 8. 6 -3

Objective 1 Solve radical equations having square root radicals. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 8. 6 -4

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. 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 © 2012, 2008, 2004 Pearson Education, Inc. Slide 8. 6 -5

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 © 2012, 2008, 2004 Pearson Education, Inc. Slide 8. 6 -6

EXAMPLE 2 Using the Squaring Property with a Radical on Each Side Solve. Solution: Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 8. 6 -7

Objective 2 Identify equations with no solutions. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 8. 6 -8

EXAMPLE 3 Using the Squaring Property When One Side Is Negative Solve. Solution: Check: False Because represents the principal or nonnegative square root of x in Example 3, we might have seen immediately that there is no solution. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 8. 6 -9

Solving a Radical Equation Step 1 Isolate a radical. Arrange the terms so that a radical is isolated on one side of the equation. Step 2 Square both sides. Step 3 Combine like terms. Step 4 Repeat Steps 1 -3 if there is still a term with a radical. Step 5 Solve the equation. Find all proposed solutions. Step 6 Check all proposed solutions in the original equation. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 8. 6 -10

EXAMPLE 4 Using the Squaring Property with a Quadratic Expression Solve Solution: Since x must be a positive number the solution set is Ø. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 8. 6 -11

Objective 3 Solve equations by squaring a binomial. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 8. 6 -12

EXAMPLE 5 Using the Squaring Property when One Side Has Two Terms Solve Solution: or Since x must be positive the solution set is {4}. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 8. 6 -13

EXAMPLE 6 Rewriting an Equation before Using the Squaring Property Solve. Solution: or The solution set is {4, 9}. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 8. 6 -14

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. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 8. 6 -15

EXAMPLE 7 Using the Squaring Property Twice Solve. Solution: The solution set is {8}. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 8. 6 -16

Objective 4 Solve radical equations having cube root radicals. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 8. 6 -17

Solve radical equations having cube root radicals. We can extend the concept of raising both sides of an equation to a power in order to solve radical equations with cube roots. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 8. 6 -18

EXAMPLE 8 Solving Equations with Cube Root Radicals Solve each equation. Solution: or Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 8. 6 -19

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