# The product of two consecutive positive integers is 120. How do you find the integers?

Jun 21, 2016

There is no such positive integer.

#### Explanation:

Let the integer be $x$. Then next integer is $x + 1$ and as their product is $120$, we have

$x \left(x + 1\right) = 120$ or

${x}^{2} + x = 120$

${x}^{2} + x - 120 = 0$

As discriminant, (${b}^{2} - 4 a c$ if the equation is $a {x}^{2} + b x + c = 0$) is

${1}^{2} - 4 \cdot 1 \cdot \left(- 120\right) = 1 + 480 = 481$ is not a perfect square, meaning thereby no rational solution,

there is no such positive integer.