Exponential Functions

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Solving Radical Equations

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Topics Solving Radical Equations

Solving Radical Equations

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 or x + 5 = - 3.
x = 3 - 5 or x = - 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 = - 2x:

()² = (- 2x)².
6x + 10 = 4x².
4x² - 6x - 10 = 0.
2(2x² - 3x - 5) = 0.
2(x + 1)(2x - 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 = - 2x is x = - 1.