n^2=n+81

Solution for n^2=n+81 equation:

n^2=n+81

We move all terms to the left:

n^2-(n+81)=0

We get rid of parentheses

n^2-n-81=0

We add all the numbers together, and all the variables

n^2-1n-81=0

a = 1; b = -1; c = -81;
Δ = b2-4ac
Δ = -12-4·1·(-81)
Δ = 325
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:

$\sqrt{\Delta}=\sqrt{325}=\sqrt{25*13}=\sqrt{25}*\sqrt{13}=5\sqrt{13}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-5\sqrt{13}}{2*1}=\frac{1-5\sqrt{13}}{2}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+5\sqrt{13}}{2*1}=\frac{1+5\sqrt{13}}{2}
$