enrotlavaec

Answered

2022-06-24

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

Answer & Explanation

humusen6p

Expert

2022-06-25Added 22 answers

The numerator factors as ${x}^{2}-2x=x\left(x-2\right)$
By the quadratic formula, the denominator has roots $2$ and $-3$. So the denominator factors as ${x}^{2}+x-6=\left(x-2\right)\left(x+3\right)$
Hence $\frac{{x}^{2}-2x}{{x}^{2}+x-6}=\frac{x\left(x-2\right)}{\left(x-2\right)\left(x+3\right)}=\frac{x}{x+3}$

crossoverman9b

Expert

2022-06-26Added 5 answers

${x}^{2}+x-6=\left(x-2\right)\left(x+3\right)\to \frac{x\left(x-2\right)}{\left(x+3\right)\left(x-2\right)}=\frac{x}{x+3}$, so your result is incorrect, as the next counterexample shows:
$x=1;\frac{{x}^{2}-2x}{{x}^{2}+x-6}=\frac{1}{4}\ne 1=\frac{3}{3x}$

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