Chapter 8 Section 6

Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. 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 © 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 Slide 8.6-3

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 1 Solve radical equations having square root radicals. Slide 8.6-4

Copyright © 2012, 2008, 2004 Pearson Education, Inc. 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. Solve radical equations having square root radicals. 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. Slide 8.6-5

Copyright © 2012, 2008, 2004 Pearson Education, Inc. 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. Slide EXAMPLE 1 Using the Squaring Property of Equality

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

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 2 Identify equations with no solutions. Slide 8.6-8

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: 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: Slide EXAMPLE 3 Using the Squaring Property When One Side Is Negative

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solving a Radical Equation Step 1Isolate a radical. Arrange the terms so that a radical is isolated on one side of the equation. Solving a Radical Equation. 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. Slide

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

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

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

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

Copyright © 2012, 2008, 2004 Pearson Education, Inc. 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 2x + 1 must be squared to get 4x 2 + 4x + 1. It is incorrect to square the 2x and the 1 separately to get 4x Slide

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

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

Copyright © 2012, 2008, 2004 Pearson Education, Inc. 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. Slide

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