n(n-1)=206

Solution for n(n-1)=206 equation:

n(n-1)=206

We move all terms to the left:

n(n-1)-(206)=0

We multiply parentheses

n^2-1n-206=0

a = 1; b = -1; c = -206;
Δ = b2-4ac
Δ = -12-4·1·(-206)
Δ = 825
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:

$\sqrt{\Delta}=\sqrt{825}=\sqrt{25*33}=\sqrt{25}*\sqrt{33}=5\sqrt{33}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-5\sqrt{33}}{2*1}=\frac{1-5\sqrt{33}}{2}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+5\sqrt{33}}{2*1}=\frac{1+5\sqrt{33}}{2}
$