Advanced Math / Nonlinear Equations One Variable

Radical and Rational Equations in One Variable

Radical and rational equations require attention to restrictions. Rational equations can have excluded denominator values, and radical equations can produce extraneous solutions after squaring.

Least you need to know

Before solving a rational equation, note any values that make a denominator equal to zero.
After squaring both sides of a radical equation, check candidate solutions in the original equation.
A radical expression such as $\sqrt{x-a}$ is always nonnegative.
Linear-looking equations with fractions can still have domain restrictions.

Key notation

$\sqrt{x+9}=x-3$ — A radical equation where the right side must be nonnegative
$\frac{x+1}{x-2}=3$ — A rational equation with excluded value $x=2$

Worked example

To solve $\sqrt{x-2}=x-4$, first note that $x-4\ge0$.
Square both sides: $x-2=(x-4)^2$.
Solve the resulting quadratic and check each candidate in the original equation.
Only candidates that satisfy the original radical equation are valid.

Common mistakes

Squaring can introduce a solution that does not satisfy the original equation.
A value that makes a denominator zero is not allowed even if it appears after algebraic manipulation.
Do not assume both quadratic roots work in a radical equation.

How to recognize it

The problem asks for the solution to an equation containing a square root.
The problem asks for a solution to an equation containing a variable in a denominator.
The problem asks which candidate solution is valid.

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