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

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Answer 1 · 2015-03-28T00:50:06

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

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

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
$ab = +24$

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

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

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