Check for extraneous solutions

A solution to a rational equation must belong to the domain of the original equation, so values that make any denominator zero are excluded. After clearing denominators or applying other transformations, a candidate value is checked in the original equation; values that are undefined or do not produce equality are extraneous and must be rejected. This distinguishes roots of a transformed equation from solutions of the original equation.

Detailed Explanation: Check for extraneous solutions

A solution to a rational equation must be in the domain of the original equation. This means no denominator can equal zero.

Consider the equation

\frac{x^2-4}{x-2}=0.

Step 1: Find excluded values

The denominator is $x-2$ . Set it equal to zero:

x-2=0 \quad \Rightarrow \quad x=2.

So $x=2$ is excluded from the domain.

Step 2: Clear the denominator

For values in the domain, multiply both sides by $x-2$ :

\frac{x^2-4}{x-2}(x-2)=0(x-2).

This gives

x^2-4=0.

Factor:

(x-2)(x+2)=0.

Therefore, the possible values are

x=2 \quad \text{or} \quad x=-2.

These are called candidates until they are checked in the original equation.

Step 3: Check each candidate in the original equation

For $x=2$ :

\frac{2^2-4}{2-2}=\frac{0}{0},

which is undefined. Thus, $x=2$ is an extraneous solution and must be rejected.

For $x=-2$ :

\frac{(-2)^2-4}{-2-2} = \frac{4-4}{-4} = 0.

This makes the original equation true, so $x=-2$ is a solution.

Therefore, the solution is

\boxed{x=-2}.

Always check candidate values in the original equation, especially values that make a denominator equal to zero.

Learn by doing: Check for extraneous solutions

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Extraneous Solutions - Rational Fraction - Potentials to Extraneous

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