Solve the Quadratic Equation by Factorization: 1/(x − 3) + 2/(x − 2) = 8/x

Question:

$$ \frac{1}{x-3}+\frac{2}{x-2}=\frac{8}{x}, \quad x\ne0,2,3 $$

Solution

$$ \frac{x-2+2(x-3)}{(x-3)(x-2)}=\frac{8}{x} $$

$$ \frac{3x-8}{(x-3)(x-2)}=\frac{8}{x} $$

$$ x(3x-8)=8(x-3)(x-2) $$

$$ 3x^2-8x=8x^2-40x+48 $$

$$ 5x^2-32x+48=0 $$

$$ 5x^2-20x-12x+48=0 $$

$$ 5x(x-4)-12(x-4)=0 $$

$$ (x-4)(5x-12)=0 $$

Either

$$ x-4=0 $$

$$ x=4 $$

or

$$ 5x-12=0 $$

$$ x=\frac{12}{5} $$

Hence,

$$ \boxed{x=4,\ \frac{12}{5}} $$