# How do you multiply  1 / (x(x - 2))+ x /(x - 2) = 10/ x?

May 15, 2015

$\frac{1}{x \left(x - 2\right)} + \frac{x}{x - 2} = \frac{10}{x}$

L.C.M of $x \left(x - 2\right) , \left(x - 2\right) \mathmr{and} x = x \left(x - 2\right)$

$\frac{1}{x \left(x - 2\right)} + \frac{x \times x}{\left(x - 2\right) \times x} = \frac{10 \times \left(x - 2\right)}{x \times \left(x - 2\right)}$

$\frac{1 + {x}^{2}}{x \left(x - 2\right)} = \frac{10 x - 20}{x \left(x - 2\right)}$

$1 + {x}^{2} = 10 x - 20$

${x}^{2} - 10 x + 21 = 0$

We can Split the Middle Term of this expression to factorise it
In this technique, if we have to factorise an expression like $a {x}^{2} + b x + c$, we need to think of 2 numbers such that:

${N}_{1} \cdot {N}_{2} = a \cdot c = 1 \times 21 = 21$

And

${N}_{1} + {N}_{2} = b = - 10$
After trying out a few numbers we get :

${N}_{1} = - 3$ and ${N}_{2} = - 7$
$- 3 \times - 7 = 21$, and $\left(- 3\right) + \left(- 7\right) = - 10$

${x}^{2} - 10 x + 21 = {x}^{2} - 3 x - 7 x + 21$

$= x \left(x - 3\right) - 7 \left(x - 3\right)$
$= \left(x - 3\right) \left(x - 7\right)$

the solutions are $x = 3 , x = 7$