Question

find all vertical asymptotes of the following function.
f(x)=\frac{2 x^{2}-7 x}{2 x^{2}+8 x-10}
answer attempt 1 out of 2
no vertical asymptotes

Explanation:

Step1: Factor numerator and denominator

Numerator: \(2x^{2}-7x = x(2x - 7)\)
Denominator: \(2x^{2}+8x - 10=2(x^{2}+4x - 5)=2(x + 5)(x - 1)\)
So \(f(x)=\frac{x(2x - 7)}{2(x + 5)(x - 1)}\)

Step2: Find values that make denominator zero

Set \(2(x + 5)(x - 1)=0\)
\(x+5 = 0\) gives \(x=-5\)
\(x - 1=0\) gives \(x = 1\)

Step3: Check if these values make numerator zero

For \(x=-5\): \(2(-5)^{2}-7(-5)=50 + 35=85
eq0\)
For \(x = 1\): \(2(1)^{2}-7(1)=2 - 7=-5
eq0\)

Answer:

The vertical asymptotes are \(x=-5\) and \(x = 1\)