Solve the Quadratic Equation by Factorization: 1/(x − 1) − 1/(x + 5) = 6/7

Question:

$$ \frac{1}{x-1}-\frac{1}{x+5}=\frac{6}{7}, \quad x\ne1,-5 $$

Solution

$$ \frac{(x+5)-(x-1)}{(x-1)(x+5)}=\frac{6}{7} $$

$$ \frac{6}{(x-1)(x+5)}=\frac{6}{7} $$

$$ (x-1)(x+5)=7 $$

$$ x^2+4x-5=7 $$

$$ x^2+4x-12=0 $$

$$ x^2+6x-2x-12=0 $$

$$ x(x+6)-2(x+6)=0 $$

$$ (x+6)(x-2)=0 $$

Either

$$ x+6=0 $$

$$ x=-6 $$

or

$$ x-2=0 $$

$$ x=2 $$

Hence,

$$ \boxed{x=-6,\ 2} $$