To solve a radical equation, perform inverse operations in the usual way. But
take note:
= | *a*|
, and thus expressions such as
must be solved as absolute value expressions for more on solving
equations containing absolute values. It is not necessary to solve
()^{2}
as an absolute value expression.

*Example*: Solve for
*x*
:
(*x* + 5)^{2} = 18
.

= .

|x+ 5| = 3 .

x+ 5 = 3 orx+ 5 = - 3 .

x= 3 - 5 orx= - 3 - 5 .

Since we cannot take the square root of a negative number, there
are often numbers which appear to be solutions but do not actually make the
equation true. For example, we get two solutions when we solve
= - 2*x*
:

()^{2} = (- 2*x*)^{2}
.

6*x* + 10 = 4*x*
^{2}
.

4*x*
^{2} - 6*x* - 10 = 0
.

2(2*x*
^{2} - 3*x* - 5) = 0
.

2(*x* + 1)(2*x* - 5) = 0
.

*x* = - 1
or
.

We can plug -1 in for
*x*
in the original equation to check that it makes the
equation true:

= - 2(- 1)
?

= 2
? Yes.

True.

However, when we plug
in for
*x*
:

= - 2()
?

= - 5
? No.

False.

is an extraneous solution, and the only solution to
= - 2*x*
is
*x* = - 1
.

As demonstrated by this example, we must check *all* "solutions" and
eliminate false solutions.

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