n(n-1)=420

Solution for n(n-1)=420 equation:

n(n-1)=420

We move all terms to the left:

n(n-1)-(420)=0

We multiply parentheses

n^2-1n-420=0

a = 1; b = -1; c = -420;
Δ = b2-4ac
Δ = -12-4·1·(-420)
Δ = 1681
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{1681}=41$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-41}{2*1}=\frac{-40}{2}
=-20
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+41}{2*1}=\frac{42}{2}
=21
$