Question

For what values of the variable is the rational expression undefined? $$\frac{x+2}{x^{2}-4}$$

Short answer

The rational expression \(\frac{x+2}{x^{2}-4}\) is undefined for \(x = 2, -2\). These values make the denominator equal to zero, which is forbidden in mathematics because division by zero is undefined.

Step-by-step solution

Identify the denominator

The denominator of the given rational expression \(\frac{x+2}{x^{2}-4}\) is \(x^{2}-4\). This will be the focus since the rational expression is undefined wherever the denominator equals zero.

Solve the equation \(x^{2}-4 = 0\)

To find the values of \(x\) that make the denominator zero, set the denominator equal to zero. Now the equation will look like this: \(x^{2}-4 = 0\). Solve this equation for \(x\).

Applying difference of squares

In order to solve the equation, you recognize that the denominator can be factored as the difference of two squares (because \(4 = 2^{2}\)). So the equation \(x^{2}-4 = 0\) can be factored as \((x-2)(x+2) = 0\)

Solve for \(x\)

By setting each factor equal to zero and solving for \(x\), you find \(x = 2\) for \((x-2) = 0\) and \(x = -2\) for \((x+2) = 0\)