Problem 6 - College Algebra

Question

Find all numbers that must be excluded from the domain of each rational expression. $$\frac{x-3}{x^{2}+4 x-45}$$

Step-by-Step Solution

Answer

The numbers that must be excluded from the domain of the given rational expression are 5 and -9.

Step 1: Identify Quadratic Expression in the Denominator

Recognize that the denominator in the expression is a quadratic expression \(x^{2}+4 x-45\).

Step 2: Factor the Quadratic Expression

By factoring the quadratic expression \(x^{2}+4 x-45\) we find: \((x-5)(x+9)\).

Step 3: Set each factor equal to zero

Solve for x by setting each factor equal to zero, thus we have: \(x-5 = 0\) and \(x+9 = 0\). After solving both equations, two values of x will be determined. Subtracting -5 from both sides in the first equation, we get \(x=5\). In the second equation, subtracting 9 from both sides gives \(x=-9\).

Step 4: Identify Excluded Values

All x-values that make the denominator 0 are to be excluded from the domain of the rational expression because division by zero is undefined. In this case, the values 5 and -9.