Question

For what values of the variable is the rational expression undefined? $$\frac{x+4}{x^{2}+x-6}$$

Short answer

The rational expression is undefined when \(x = 2\) or \(x = -3\).

Step-by-step solution

Identify the Denominator

The denominator of the given rational expression is \(x^{2} + x - 6\). This is the part of the expression we are interested in, because if it equals zero, the whole expression becomes undefined.

Equate Denominator to Zero and Solve

Now, to find the values for which the denominator equals zero, set the denominator equal to zero and solve for \(x\). \n So, \(x^{2} + x - 6 = 0\). This is a quadratic equation, for which solutions can be found either by factoring, completing the square, or using the quadratic formula. Here it can be factored into \((x-2)(x+3) = 0\). By setting each factor equal to zero and solving, we get \(x = 2\) and \(x = -3\)

Summarize the Results

The rational expression is undefined for \(x = 2\) and \(x = -3\). These are the values that make the denominator zero, leading to division by zero, which is undefined.