Express \( \frac{5x-12}{(x-2)(x-3)} \) in partial fractions

5x−12(x−2)(x−3)=Ax−2+Bx−3 $\frac{5x-12}{(x-2)(x-3)}=\frac{A}{x-2}+\frac{B}{x-3}$

= A(x−3)+B(x−2)(x−2)(x−3) $\frac{A(x-3)+B(x-2)}{(x-2)(x-3)}$

⟹5x−12=Ax−3A+Bx−2B $⟹5x-12=Ax-3A+Bx-2B$

A+B=5...(i) $A+B=5...(i)$

−(3A+2B)=−12⟹3A+2B=12...(ii) $-(3A+2B)=-12⟹3A+2B=12...(ii)$

From (i), A=5−B $A=5-B$

3(5−B)+2B=12 $3(5-B)+2B=12$

15−3B+2B=12⟹B=3 $15-3B+2B=12⟹B=3$

A+3=5⟹A=2 $A+3=5⟹A=2$

5x−12(x−2)(x−3)=2x−2+3x−3