# How do you solve 5 / (x - 2)^2 = 10 / (3x + 3)?

Aug 29, 2016

$x = 0.5 \mathmr{and} x = 5$

#### Explanation:

$\frac{5}{x - 2} ^ 2 = \frac{10}{3 x + 3}$

$15 x + 15 = 10 \left({x}^{2} - 4 x + 4\right) = 10 {x}^{2} - 40 x + 40$

$\therefore 10 {x}^{2} - 55 x + 25 = 0$

$5 \left(2 {x}^{2} - 11 x + 5\right) = 0$

$5 \left(2 x - 1\right) \left(x - 5\right) = 0$

$\left(2 x - 1\right) \left(x - 5\right) = 0$

graph{(y-5/(x-2)^2)(y-10/(3*x+3))=0 [-1, 6, -1, 3]}

graph{10x^2-55x+25 [-1, 6, -60, 10]}

Aug 29, 2016

$x = \frac{1}{2} \mathmr{and} x = 5$

#### Explanation:

There is one term (a fraction) on each side of the equation.
We can cross-multiply to get rid of the denominators.

$10 {\left(x - 2\right)}^{2} = 5 \left(3 x + 3\right) \textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots} \div 5$

$2 {\left(x - 2\right)}^{2} = \left(3 x + 3\right)$

$2 \left({x}^{2} - 4 x + 4\right) = 3 x + 3$

$2 {x}^{2} - 8 x + 8 - 3 x - 3 = 0$

$2 {x}^{2} - 11 x + 5 = 0$

Find factors of 2 and 5 which add up to 11.
(Work towards $1 x + 10 x = 11 x$)

$\left(2 x - 1\right) \left(1 - 5\right) = 0$

$x = \frac{1}{2} \mathmr{and} x = 5$