Skill: 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.