# How do you multiply (n^2)/(2)-(5n)/(6) -2 =0?

May 9, 2015

${n}^{2} / 2$ $- \frac{5 n}{6}$ $- 2 = 0$

Here the L.C.M = 6

$\frac{{n}^{2} \times 3}{2 \times 3}$ $- \frac{5 n}{6}$ $- \frac{2 \times 6}{1 \times 6} = 0$

$\frac{3 {n}^{2}}{6}$ $- \frac{5 n}{6}$ $- \frac{12}{6} = 0$
$\frac{3 {n}^{2} - 5 n - 12}{6} = 0$

$3 {n}^{2} - 5 n - 12 = 6 \times 0$

$3 {n}^{2} - 5 n - 12 = 0$

factoring by grouping / splitting the middle term:
$3 {n}^{2} - 5 n - 12 = 0$

($3 \times - 12 = - 36$ , $- 36 = - 9 \times 4$ and $- 5$ = $4 - 9$)

so,
$3 {n}^{2} - 9 n + 4 n - 12 = 0$

taking out the common terms:
$3 n \left(n - 3\right) + 4 \left(n - 3\right) = 0$
on grouping:
$\left(3 n + 4\right) \left(n - 3\right) = 0$

here, $n$ has two solutions :
$n = 3 \mathmr{and} n = - \frac{4}{3}$