Solving Radical Equations Section 9.5 Solving Radical Equations

Objectives Solve equations containing one radical Solve equations containing two radicals Solve formulas containing radicals

Objective 1: Solve Equations Containing One Radical Radical equations contain a radical expression with a variable in the radicand. Some examples are To solve equations containing radicals, we will use the power rule. The Power Rule: If we raise two equal quantities to the same power, the results are equal quantities. If x, y, and n are real numbers and x = y, then for any exponent n. The Square of a Square Root: For any nonnegative real number a, .

Objective 1: Solve Equations Containing One Radical Solving an Equation Containing Radicals: Isolate a radical term on one side of the equation. Raise both sides of the equation to the power that is the same as the index of the radical. If it still contains a radical, go back to step 1. If it does not contain a radical, solve the resulting equation. Check the proposed solutions in the original equation.

EXAMPLE 1 Solve: Strategy We will use the power rule and square both sides of the equation. Why Squaring both sides will produce, on the left side, the expression that simplifies to . This step clears the equation of the radical.

EXAMPLE 1 Solution Solve: We must check the proposed solution of 13 to see whether it satisfies the original equation. Since 13 satisfies the original equation, it is the solution. The solution set is {13}.

Objective 2: Solve Equations Containing Two Radicals To solve an equation containing two radicals, we want to have one radical on the left side and one radical on the right side. When more than one radical appears in an equation, we must often use the power rule more than once.

EXAMPLE 7 Find: Strategy We will square both sides to clear the equation of both radicals. Why We can immediately square both sides since each radical is isolated on one side of the equation.

EXAMPLE 7 Find: Solution

EXAMPLE 7 Find: Check: We check the solution by substituting –1 for x in the original equation. The solution is –1 and the solution set is {–1}.

Objective 3: Solve Formulas Containing Radicals To solve a formula for a variable means to isolate that variable on one side of the equation, with all other quantities on the other side.

Depreciation Rates. Some office equipment that is now worth V dollars originally cost C dollars 3 years ago. The rate r at which it has depreciated is given by . Solve the formula for C. EXAMPLE 9 Strategy To isolate the radical, we will subtract 1 from both sides. We can then eliminate the radical by cubing both sides. Why Cubing both sides will produce, on the right, the expression that simplifies to . This step clears the equation of the radical.

Depreciation Rates. Some office equipment that is now worth V dollars originally cost C dollars 3 years ago. The rate r at which it has depreciated is given by EXAMPLE 9 Solution We begin by isolating the cube root on the right side of the equation. . Solve the formula for C.