108=n(n-2)-180,n

Solution for 108=n(n-2)-180,n equation:

108=n(n-2)-180.n

We move all terms to the left:

108-(n(n-2)-180.n)=0


We calculate terms in parentheses: -(n(n-2)-180.n), so:

n(n-2)-180.n

We add all the numbers together, and all the variables

-180n+n(n-2)

We multiply parentheses

n^2-180n-2n

We add all the numbers together, and all the variables

n^2-182n

Back to the equation:

-(n^2-182n)


We get rid of parentheses

-n^2+182n+108=0

We add all the numbers together, and all the variables

-1n^2+182n+108=0

a = -1; b = 182; c = +108;
Δ = b2-4ac
Δ = 1822-4·(-1)·108
Δ = 33556
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:

$\sqrt{\Delta}=\sqrt{33556}=\sqrt{4*8389}=\sqrt{4}*\sqrt{8389}=2\sqrt{8389}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(182)-2\sqrt{8389}}{2*-1}=\frac{-182-2\sqrt{8389}}{-2}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(182)+2\sqrt{8389}}{2*-1}=\frac{-182+2\sqrt{8389}}{-2}
$