# The product of two consecutive positive integers is 11 more than their sum, what are the integers?

May 29, 2015

If the integers are $m$ and $m + 1$, then we are given:

$m \times \left(m + 1\right) = m + \left(m + 1\right) + 11$

That is:

${m}^{2} + m = 2 m + 12$

Subtract $2 m + 12$ from both sides to get:

$0 = {m}^{2} - m - 12 = \left(m - 4\right) \left(m + 3\right)$

This equation has solutions $m = - 3$ and $m = 4$

We were told that $m$ and $m + 1$ are positive, so we can reject $m = - 3$, leaving the unique solution $m = 4$.

So the integers are $m = 4$ and $m + 1 = 5$.