512=n^2

Solution for 512=n^2 equation:

512=n^2

We move all terms to the left:

512-(n^2)=0

We add all the numbers together, and all the variables

-1n^2+512=0

a = -1; b = 0; c = +512;
Δ = b2-4ac
Δ = 02-4·(-1)·512
Δ = 2048
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{2048}=\sqrt{1024*2}=\sqrt{1024}*\sqrt{2}=32\sqrt{2}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-32\sqrt{2}}{2*-1}=\frac{0-32\sqrt{2}}{-2}
=-\frac{32\sqrt{2}}{-2}
=-\frac{16\sqrt{2}}{-1}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+32\sqrt{2}}{2*-1}=\frac{0+32\sqrt{2}}{-2}
=\frac{32\sqrt{2}}{-2}
=\frac{16\sqrt{2}}{-1}
$