n(n-2)=420

Solution for n(n-2)=420 equation:

n(n-2)=420

We move all terms to the left:

n(n-2)-(420)=0

We multiply parentheses

n^2-2n-420=0

a = 1; b = -2; c = -420;
Δ = b2-4ac
Δ = -22-4·1·(-420)
Δ = 1684
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:

$\sqrt{\Delta}=\sqrt{1684}=\sqrt{4*421}=\sqrt{4}*\sqrt{421}=2\sqrt{421}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2\sqrt{421}}{2*1}=\frac{2-2\sqrt{421}}{2}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{421}}{2*1}=\frac{2+2\sqrt{421}}{2}
$