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

Question:

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

Solution

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

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

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

$$ 3x^2-6x+2=0 $$

$$ 3x^2-3x-3x+2=0 $$

$$ 3x(x-1)-2(x-1)=0 $$

$$ (3x-2)(x-1)=0 $$

Either

$$ 3x-2=0 $$

$$ x=\frac{2}{3} $$

or

$$ x-1=0 $$

$$ x=1 $$

Hence,

$$ \boxed{x=\frac{2}{3},\ 1} $$