(n(n-2))/2=54

Solution for (n(n-2))/2=54 equation:

(n(n-2))/2=54

We move all terms to the left:

(n(n-2))/2-(54)=0

We multiply all the terms by the denominator


(n(n-2))-54*2=0


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

n(n-2)

We multiply parentheses

n^2-2n

Back to the equation:

+(n^2-2n)


We add all the numbers together, and all the variables

(n^2-2n)-108=0

We get rid of parentheses

n^2-2n-108=0

a = 1; b = -2; c = -108;
Δ = b2-4ac
Δ = -22-4·1·(-108)
Δ = 436
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:

$\sqrt{\Delta}=\sqrt{436}=\sqrt{4*109}=\sqrt{4}*\sqrt{109}=2\sqrt{109}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2\sqrt{109}}{2*1}=\frac{2-2\sqrt{109}}{2}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{109}}{2*1}=\frac{2+2\sqrt{109}}{2}
$