How can I solve this partial fraction #(7x-10)/(4x^2-12x+9)#?

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Answer 1 · 2018-08-01T23:29:28

#(7x-10)/(4x^2-12x+9) = 7/(2(2x-3))+1/(2(2x-3)^2)#

Given:

#(7x-10)/(4x^2-12x+9)#

First note that:

#4x^2-12x+9 = (2x-3)^2#

is a perfect square trinomial.

So we are looking for a partial fraction expansion of the form:

#(7x-10)/(4x^2-12x+9) = A/(2x-3)+B/(2x-3)^2#

Mutiplying both sides by #(2x-3)^2#, this becomes:

#7x-10 = A(2x-3)+B#

Looking at the coefficient of #x#, we can deduce that:

#A=7/2#

Putting #x=3/2#, we find:

#B = 7(color(blue)(3/2))-10 = 1/2#

So:

#(7x-10)/(4x^2-12x+9) = 7/(2(2x-3))+1/(2(2x-3)^2)#