# How do solve n^2 - 10n + 22 = -2?

Mar 28, 2015

${n}^{2} - 10 n + 22 = - 2$
is equivalent to
${n}^{2} - 10 n + 24 = 0$

We would like to factor the left hand size to get something of the form
$\left(n + a\right) \left(n + b\right) = {n}^{2} - 10 n + 24$

Since the final term is positive, $a$ and $b$ must have the same sign
and
since the coefficient of $x$ is negative they must both be negative.

So we are looking for negative values $a$ and $b$ such that
$a + b = - 10$
and
$a b = + 24$

Assuming integer solutions there are very few possibilities.
By trial and error we can derive
$\left(n - 6\right) \left(n - 4\right) = {n}^{2} - 10 n + 24 = 0$

which implies that
$n = 6$ or $n = 4$