12^2+13^2=x^2

Solution for 12^2+13^2=x^2 equation:

12^2+13^2=x^2

We move all terms to the left:

12^2+13^2-(x^2)=0

We add all the numbers together, and all the variables

-1x^2+313=0

a = -1; b = 0; c = +313;
Δ = b2-4ac
Δ = 02-4·(-1)·313
Δ = 1252
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{1252}=\sqrt{4*313}=\sqrt{4}*\sqrt{313}=2\sqrt{313}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-2\sqrt{313}}{2*-1}=\frac{0-2\sqrt{313}}{-2}
=-\frac{2\sqrt{313}}{-2}
=-\frac{\sqrt{313}}{-1}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+2\sqrt{313}}{2*-1}=\frac{0+2\sqrt{313}}{-2}
=\frac{2\sqrt{313}}{-2}
=\frac{\sqrt{313}}{-1}
$