13^2=n^2+5

Solution for 13^2=n^2+5 equation:

13^2=n^2+5

We move all terms to the left:

13^2-(n^2+5)=0

We add all the numbers together, and all the variables

-(n^2+5)+169=0

We get rid of parentheses

-n^2-5+169=0

We add all the numbers together, and all the variables

-1n^2+164=0

a = -1; b = 0; c = +164;
Δ = b2-4ac
Δ = 02-4·(-1)·164
Δ = 656
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:

$\sqrt{\Delta}=\sqrt{656}=\sqrt{16*41}=\sqrt{16}*\sqrt{41}=4\sqrt{41}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-4\sqrt{41}}{2*-1}=\frac{0-4\sqrt{41}}{-2}
=-\frac{4\sqrt{41}}{-2}
=-\frac{2\sqrt{41}}{-1}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+4\sqrt{41}}{2*-1}=\frac{0+4\sqrt{41}}{-2}
=\frac{4\sqrt{41}}{-2}
=\frac{2\sqrt{41}}{-1}
$