Question

partial fraction decomposition
3x-4/x^2-4x-12

Answer 1

You have:
\[\frac{3x-4}{x^2-4x-12}\]
Factor the bottom:
\[\frac{3x-4}{(x-6)(x+2)}\]
You have two linear factors. So you are looking for something in the form of:
\[\frac{A}{x-6}+\frac{B}{x+2}\]
So get a common denominator and set the numerators equal giving:
\[3x-4=A(x+2)+B(x-6)\]
Multiply and collect like terms:
\[3x-4=Ax+Bx+2A-6B\]
So you know that A+B=3 2A-6B=-4
B=3-A
2A-6(3-A)=-4
2A-18+6A=-4
8A=14
A=7/4
So B=3-(7/4)=5/4
So your decomposition is:
\[\frac{7}{4(x-6)}+\frac{5}{4(x+2)}\]