158 CHAPTER 1 Equations and Inequalities Equations with Radicals To solve an equation such as x - 215 - 2x = 0, in which the variable appears in a radicand, we use the following power property to eliminate the radical. Power Property If P and Q are algebraic expressions, then every solution of the equation P = Q is also a solution of the equation Pn = Qn, for any positive integer n. When the power property is used to solve equations, the new equation may have more solutions than the original equation. For example, the equation x = -2 has solution set 5-26. If we square each side of the equation x = -2, we obtain the new equation x2 = 4, which has solution set 5-2, 26. Because the solution sets are not equal, the equations are not equivalent. When we use the power property to solve an equation, it is essential to check all proposed solutions in the original equation. CAUTION Be very careful when using the power property. It does not say that the equations P = Q and Pn = Qn are equivalent. It says only that each solution of the original equation P = Q is also a solution of the new equation Pn = Qn. Solving an Equation Involving Radicals Step 1 Isolate the radical on one side of the equation. Step 2 Raise each side of the equation to a power that is the same as the index of the radical so that the radical is eliminated. If the equation still contains a radical, repeat Steps 1 and 2. Step 3 Solve the resulting equation. Step 4 Check each proposed solution in the original equation. EXAMPLE 4 Solving an Equation Containing a Radical (Square Root) Solve x - 215 - 2x = 0. SOLUTION x - 215 - 2x = 0 Step 1 x = 215 - 2x Isolate the radical. Step 2 x2 = A 215 - 2x B 2 Square each side. x2 = 15 - 2x A 2a B 2 = a, for a Ú 0.
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