Question

find all vertical asymptotes of the following function.

$f(x)=\frac{5 x^{2}-12 x-9}{2 x^{2}-16 x+30}$

no vertical asymptotes
one vertical asymptote
two vertical asymptotes
no vertical asymptotes

Explanation:

Step1: Factor numerator and denominator

• Numerator: \(5x^{2}-12x - 9=(5x + 3)(x-3)\)
• Denominator: \(2x^{2}-16x + 30 = 2(x^{2}-8x + 15)=2(x - 3)(x - 5)\)

Step2: Simplify the function

\(f(x)=\frac{(5x + 3)(x - 3)}{2(x - 3)(x - 5)}=\frac{5x+3}{2(x - 5)},x
eq3\)

Step3: Find vertical asymptotes

Vertical asymptotes occur where the denominator is zero (after simplification). Set \(2(x - 5)=0\), solve for \(x\). We get \(x = 5\)

Answer:

One Vertical Asymptote