462=n(n+1)

Solution for 462=n(n+1) equation:

462=n(n+1)

We move all terms to the left:

462-(n(n+1))=0


We calculate terms in parentheses: -(n(n+1)), so:

n(n+1)

We multiply parentheses

n^2+n

Back to the equation:

-(n^2+n)


We get rid of parentheses

-n^2-n+462=0

We add all the numbers together, and all the variables

-1n^2-1n+462=0

a = -1; b = -1; c = +462;
Δ = b2-4ac
Δ = -12-4·(-1)·462
Δ = 1849
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{1849}=43$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-43}{2*-1}=\frac{-42}{-2}
=+21
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+43}{2*-1}=\frac{44}{-2}
=-22
$