Question

The sum of a positive integer and its square is 90. What is the number?

Answer 1

`9`

Explanation:

Let `n` be the integer in question. Then we have

`n^2+n = 90`

`=> n^2+n-90 = 0`

We now have a quadratic equation to solve. We could use the quadratic formula, however we know that `n` is an integer, so instead let's try to solve by factoring instead.

`n^2+n-90 = 0`

`=> n^2 + 10n - 9n - 90 = 0`

`=> n(n+10)-9(n+10) = 0`

`=> (n-9)(n+10) = 0`

`=> n-9 = 0 or n+10 = 0`

`=> n=9 or n=-10`

As it is given that `n>0`, we can disregard the possibility that `n=-10`, leaving us with our final answer of `n=9`

Checking our result, we find that it satisfies the given conditions:

`9+9^2 = 9+81 = 90`