Alg5.1.17

Resolve $\frac{4x^2-12x+13}{4x^2-16x+15}\,$ into partial fractions.

Since the powers in the numerator and denominators are equal,

$\frac{4x^2-12x+13}{4x^2-16x+15}=\frac{4x^2-12x+13}{4x^2-6x-10x+15}=\frac{4x^2-12x+13}{(2x-3)(2x-5)}=k+\frac{A}{2x-3}+\frac{B}{2x-5}\,$

Equating the numerators,we get $4x^2-12x+13=k(2x-3)(2x-5)+A(2x-5)+B(2x-3)\,$

Substituting the values $x=\frac{3}{2},\frac{5}{2}\,$ we get

$4\cdot\frac{9}{4}-18+13=-2A,A=-2\,$

$4\cdot\frac{25}{4}-30+13=2B,B=4\,$

Comparing the coefficients of $x^2\,$ we get $4=4k,k=1\,$

Therefore, $\frac{4x^2-12x+13}{4x^2-16x+15}=1+\frac{-2}{2x-3}-\frac{4}{2x-5}\,$