6.6 Solve Radical Equations

VOCABULARY

Radical equation

An equation with a radical that has variables in the radicand

SOLVING RADICAL EQUATIONS

To solve a radical equation, follow these steps:

Step 1 _Isolate_ the radical on one side of the equation, if necessary.

Step 2 Raise each side of the equation to the same _power_ to eliminate the radical and obtain a linear, quadratic, or other polynomial equation.

Step 3 _Solve_ the polynomial equation using techniques you learned in previous chapters. Check your solution.

Example 1

Solve a radical equation

Solve

	Write original equation.
= _3^2	Square each side to eliminate the radical.
x+ 6 = _9_	Simplify.
_x_ = _3_	Subtract _6_ from each side.
The solution is _3_. Check this in the original equation

.

Solve the equation. Check your solution.

-3

Example 2

Solve an equation with a rational exponent

(3x + 4)^2/^3 = 16	Original equation
[(3x+ 4)^2/3]^3/2 = 16^3/2	Raise each side to the Power
3x + 4 = (16^1/2)^3	Apply properties of exponents.
3x+ 4 = 64	Simplify.
3x = 60	Subtract _4_ from each side.
_x_ = 20	Divide each side by _3_.
The solution is _20_. Check this in the original equation

.

Example 3

Solve an equation with an extraneous solution

x -2 =	Original equation
(x- 2)^2 = ( )^2	Square each side.
x^2 - 4x+ 4 = x+ 10	Expand left side and simplify right side.
x^2 - 5x - 6 = 0	Write in standard form.
(x- 6)(x + 1) = 0	Factor.
x- 6 = 0 or x+ 1 = 0	Zero product property
x = _6_ or x = -1	Solve for x

CHECK
Check x = _6_.	Check x = _-1_.
x -2 =	x -2 =

__4__=__4__

_-3 ¹ 3_

The only solution is __4__ (The apparent solution __- 1__ is extraneous.)

Example 4

Solve an equation with two radicals

Solve

	Write original equation.
	Square each side.
	Expand left side and simplify right side.
	Isolate radical expression.
	Divide each side by 4.
	Square each side again.
x + 6 = x^2	Simplify.
0 = x^2 - x - 6	Write in standard form.
0 = (x - 3)(x + 2)	Factor.
x - 3 = 0 or x + 2 = 0	Zero product property
x = 3 or x = -2	Solve for x
CHECK Check x = 3 .
5 = 1
Check x = - _-2 .
_4__=__4_

The only solution is -2_. (The apparent solution _3_ is extraneous.)