Problem 9 - College Algebra

Question

Simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. $$\frac{x^{2}-12 x+36}{4 x-24}$$

Step-by-Step Solution

Answer

The simplified form of the given rational expression is \(\frac{(x-6)}{4}\) and the value that should be excluded from the domain is 6.

Step 1: Factorize the Expression

Factorize the expressions in the numerator and the denominator. In the numerator, we have a perfect square trinomial, so \(x^{2}-12 x+36\) becomes \((x-6)^2\). In the denominator, 4x-24 can be re-written as 4(x-6).

Step 2: Simplify the Expression

Simplify the expression by canceling out the common expressions in the numerator and the denominator. So, \(\frac{(x-6)^2}{4(x-6)}\) simplifies to \(\frac{(x-6)}{4}\), after canceling out one of the \((x-6)\) factors.

Step 3: Find the Excluded Values

Since division by zero leads to undefined expressions, find the values of x that make the denominator zero. So, solve the equation 4(x-6) = 0, we get x = 6. This value must be excluded from the domain of the rational expression because it makes the denominator zero.