Problem 6 - College Algebra Essentials

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

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

Step 1: Equation Formation

First, set the denominator equal to zero and solve for \(x\). \(x^{2}+4 x-45 = 0\)

Step 2: Solve the Quadratic Equation

Next, we can solve the quadratic equation. The equation can be factored into (x-5)(x+9)=0.

Step 3: Find the X values

So, \(x\) can be 5 or -9. To solve, we set each factor equal to zero and solve. \(x-5 = 0 \Rightarrow x = 5\), \(x+9 = 0 \Rightarrow x = -9\) .

Step 4: Identify the Excluded Values

The 'x' values that cause the denominator to equal zero, and therefore make the function undefined, are \(x = 5\) and \(x = -9\). These values should be excluded from the domain.