Why Partial Fractions Decomposition works

Question

Consider this partial fraction. $$\dfrac{x-25}{x^2+5x-24}=\dfrac{A}{x-3}+\dfrac{B}{x+8}$$ Multiply both sides by the quadratic $x^2+5x-24$. $$x-25=A(x+8)+B(x-3)$$ From here, I've seen many textbooks set $x=3$, solve for $A$, then set $x=-8$ and solve for $B$. But how come we can do that if we assumed $x\neq3$ and $x\neq-8$ when dividing the quadratic?

Answer 1

You can use it to find A and B since you have transformed your equations, and the linear terms in x are no longer in the denominator, so you can treat the newly obtained equation as a separate new equation and thus disregard the assumptions of the previous equation. In multiplying the equation by $ x²+5x-24 $, you have slightly changed the working of the equation; the newly obtained equation works for all real x, including -8 and 3.