Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 15 Roots and Radicals

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Solving Equations Containing Radicals

Martin-Gay, Developmental Mathematics, 2e 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The Squaring Property of Equality If a = b, then a 2 = b 2.

Martin-Gay, Developmental Mathematics, 2e 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. If both sides of an equation are squared, solutions of the new equation contain all the solutions of the original equation, but might also contain additional solutions. A proposed solution of the new equation that is NOT a solution of the original equation is an extraneous solution. For this reason, we must always check the proposed solutions of radical equations in the original equation. Extraneous Solutions

Martin-Gay, Developmental Mathematics, 2e 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Helpful Hint Don’t forget to check the proposed solutions of radical equations in the original equation.

Martin-Gay, Developmental Mathematics, 2e 66 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Solve the following radical equation. True Check: Substitute into the original equation. So the solution is x = 24. Example

Martin-Gay, Developmental Mathematics, 2e 77 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Solve the following radical equation. Does NOT check, since the left side of the equation is asking for the principal square root. So this equation has no solution. Example Check: Substitute into the original equation.

Martin-Gay, Developmental Mathematics, 2e 88 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To Solve a Radical Equation Containing Square Roots Step 1: Arrange terms so that one radical is by itself on one side of the equation. That is, isolate a radical. Step 2: Square both sides of the equation. Step 3: Simplify both sides of the equation. Step 4: If equation still contains a radical, repeat Steps 1 through 3. Step 5: Solve the equation. Step 6: Check all solutions in the original equation for extraneous solutions Solving Radical Equations

Martin-Gay, Developmental Mathematics, 2e 99 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Solve: True So the solution is x = 2. Example Check: Substitute into the original equation.

Martin-Gay, Developmental Mathematics, 2e 10 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Solve: Example continued

Martin-Gay, Developmental Mathematics, 2e 11 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Substitute the value for x into the original equation, to check the solution. True False 21/4 is an extraneous solution. The only solution is x = 3. continued

Martin-Gay, Developmental Mathematics, 2e 12 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Solve: Example continued

Martin-Gay, Developmental Mathematics, 2e 13 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Substitute the value for x into the original equation, to check the solution. False This equation has no solution. continued

Martin-Gay, Developmental Mathematics, 2e 14 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Solve: Example continued

Martin-Gay, Developmental Mathematics, 2e 15 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Substitute the value for x into the original equation, to check the solution. True The solutions are x = 4 or x = 20. continued