# How do you multiply ((r+5)/(r^2-2r)) - 1= 1/(r^2-2r)?

May 21, 2015

$\left(\frac{r + 5}{{r}^{2} - 2 r}\right) - 1 = \frac{1}{{r}^{2} - 2 r}$

$\left(\frac{r + 5}{{r}^{2} - 2 r}\right) - 1 \times \left(\frac{{r}^{2} - 2 r}{{r}^{2} - 2 r}\right) = \frac{1}{{r}^{2} - 2 r}$

$\left(\frac{r + 5}{{r}^{2} - 2 r}\right) - \left(\frac{{r}^{2} - 2 r}{{r}^{2} - 2 r}\right) = \frac{1}{{r}^{2} - 2 r}$

$\left(\frac{r + 5 - {r}^{2} + 2 r}{\cancel{{r}^{2} - 2 r}}\right) = \frac{1}{\cancel{{r}^{2} - 2 r}}$

$\left(r + 5 - {r}^{2} + 2 r\right) = 1$
$- {r}^{2} + 3 r + 4 = 0$

${r}^{2} - 3 r - 4 = 0$
Factorising by splitting the middle term:

${r}^{2} - 4 r + r - 4 = 0$
$r \left(r - 4\right) + 1 \left(r - 4\right) = 0$
$\left(r + 1\right) \left(r - 4\right) = 0$

color(green)(r = -1,4 are the two solutions