The sum of a number and its square is 72. what is the number?

Any equation that has something squared in it always has two solutions.
The equation that you'd use to solve this problem is no exception.

You said that    x² + x = 72

Subtract  72  from each side, and it says      x² + x - 72 = 0

That's the equation to solve, and when you do, the value of 'x'
is the answer to your puzzler.

You can solve the equation using the quadratic formula, or even
by factoring if you can handle it that way.

       (x + 9) (x - 8) = 0

         x = -9
and
         x = 8 .

Sure enough, they both work:

x = -9
x² = 81
x² + x = 81 - 9 = 72

x = 8
x² = 64
x² + x = 64 + 8 = 72

         yay !