Solve quadratic fraction

Question

I would like to simplify the fraction $$\frac{x^2-2x}{x^2+x-6}$$ I know from Mathematica that it should equal $\frac3{3x}$ but how do I get there?

Answer 1

The numerator factors as $x^{2}-2x=x(x-2)$.

By the quadratic formula, the denominator has roots $2$ and $-3$. So the denominator factors as $x^{2}+x-6=(x-2)(x+3)$.

Hence $\frac{x^{2}-2x}{x^{2}+x-6}=\frac{x(x-2)}{(x-2)(x+3)}=\frac{x}{x+3}$.

Answer 2

$x^2+x-6 = (x-2)(x+3)\rightarrow \frac{x(x-2)}{(x+3)(x-2)}=\frac{x}{x+3}$, so your result is incorrect, as the next counterexample shows:

$x=1; \frac{x^2-2x}{x^2+x-6}=\frac{1}{4}\neq 1=\frac{3}{3x}$