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Chapter 16

Reining in Radical and Absolute Value Equations

Aradical equation contains at least one term that’s a square root, cube root, or some other root. When solving radical equations, you apply a method that’s effective but comes with a built-in error possibility; you may find (and need to recognize) extraneous solutions. You need to rewrite absolute value equations to solve them. The solutions of the rewrites are then the solutions of the original equation.

The Problems You’ll Work On

Here’s just a sampling of the radical things you work on in this chapter:

 Rewriting equations with only two terms and squaring both sides to solve

 Squaring both sides of an equation when starting with three terms

 Dealing with more than one radical term

 Catching extraneous solutions

 Graphing absolute value statements for clarity

 Solving absolute value equations after writing corresponding equations

 Checking for nonsense answers

What to Watch Out For

Here are a few things that may rock your boat, so be on the lookout:

 Squaring binomials correctly and not forgetting the middle term

 Factoring statements containing radicals correctly

 Checking for extraneous solutions by using the original equation, not the version changed in the process

 Catching impossible situations in initial absolute value equations

Solving Basic Radical Equations

686–689 Solve each radical equation by squaring both sides.

686.

687.

688.

689.

Checking for Extraneous Roots

690–697 Solve the radical equations by squaring both sides; check for extraneous solutions.

690.

691.

692.

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695.

696.

697.

Squaring Both Sides of Equations Twice

698–701 Solve each radical equation by squaring both sides of the equation twice.

698.

699.

700.

701.

Solving Radicals with Roots Other Than Square Roots

702–703 Solve the radical equations.

702.

703.

Solving Absolute Value Equations

704–713 Solve each absolute value equation by writing the two corresponding linear equations and solving.

704.

705.

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713.

Handling Absolute Value Equations with No Solution

714–715 Solve the absolute value equations, and check the answers carefully.

714.

715.