462=n(n+1)

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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 $

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