Question

question
find all vertical asymptotes of the following function.
$f(x) = \frac{3x^2 - 6x - 144}{2x^2 - 5x}$
answer attempt 1 out of 5
no vertical asymptotes

Explanation:

Step1: Factor numerator and denominator

Factor numerator: \(3x^2 - 6x - 144 = 3(x^2 - 2x - 48)=3(x - 8)(x + 6)\)
Factor denominator: \(2x^2 - 5x = x(2x - 5)\)

Step2: Find values that make denominator zero (and not numerator)

Denominator zero when \(x(2x - 5)=0\), so \(x = 0\) or \(2x - 5 = 0\Rightarrow x=\frac{5}{2}\).
Check numerator at these \(x\):

• At \(x = 0\): \(3(0 - 8)(0 + 6)=-144

eq0\)

• At \(x=\frac{5}{2}\): \(3(\frac{5}{2}-8)(\frac{5}{2}+6)=3(-\frac{11}{2})(\frac{17}{2})

eq0\)

Answer:

\(x = 0\) and \(x=\frac{5}{2}\)