90=(n^2-81)

Solution for 90=(n^2-81) equation:

90=(n^2-81)

We move all terms to the left:

90-((n^2-81))=0


We calculate terms in parentheses: -((n^2-81)), so:

(n^2-81)

We get rid of parentheses

n^2-81

Back to the equation:

-(n^2-81)


We get rid of parentheses

-n^2+81+90=0

We add all the numbers together, and all the variables

-1n^2+171=0

a = -1; b = 0; c = +171;
Δ = b2-4ac
Δ = 02-4·(-1)·171
Δ = 684
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:

$\sqrt{\Delta}=\sqrt{684}=\sqrt{36*19}=\sqrt{36}*\sqrt{19}=6\sqrt{19}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-6\sqrt{19}}{2*-1}=\frac{0-6\sqrt{19}}{-2}
=-\frac{6\sqrt{19}}{-2}
=-\frac{3\sqrt{19}}{-1}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+6\sqrt{19}}{2*-1}=\frac{0+6\sqrt{19}}{-2}
=\frac{6\sqrt{19}}{-2}
=\frac{3\sqrt{19}}{-1}
$