Advanced Math / Nonlinear Equations One Variable

Radical and Rational Equations

Radical and rational equations are common sources of errors because restrictions and extraneous solutions must be checked. Squaring and clearing denominators can introduce invalid candidates.

Least you need to know

Check radical equation solutions in the original equation.
Values that make a denominator zero are never allowed.
When squaring both sides, solve the resulting equation and then verify each candidate.
A square root is always nonnegative.

Key notation

$\sqrt{x+4}=x-2$ — A radical equation that may produce an extraneous solution
$\frac{6}{x-1}=3$ — A rational equation with excluded value $x=1$

Worked example

To solve $\sqrt{x+4}=x-2$, square both sides to get $x+4=(x-2)^2$.
Solve the resulting quadratic.
Check each candidate in the original equation because squaring can introduce extraneous solutions.

Common mistakes

A candidate that satisfies the squared equation may fail the original equation.
Do not multiply by a denominator without noting excluded values.
The right side of a radical equation must be nonnegative if it equals a principal square root.

How to recognize it

The variable appears under a radical.
The variable appears in a denominator.
The problem asks for a valid solution or an excluded value.

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Radical and Rational Equations in One Variable