3n(n+1)/(n-1)=126

Solution for 3n(n+1)/(n-1)=126 equation:

3n(n+1)/(n-1)=126

We move all terms to the left:

3n(n+1)/(n-1)-(126)=0


Domain of the equation: (n-1)!=0

We move all terms containing n to the left, all other terms to the right

n!=1

n∈R


We multiply all the terms by the denominator


3n(n+1)-126*(n-1)=0

We multiply parentheses

3n^2+3n-126n+126=0

We add all the numbers together, and all the variables

3n^2-123n+126=0

a = 3; b = -123; c = +126;
Δ = b2-4ac
Δ = -1232-4·3·126
Δ = 13617
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:

$\sqrt{\Delta}=\sqrt{13617}=\sqrt{9*1513}=\sqrt{9}*\sqrt{1513}=3\sqrt{1513}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-123)-3\sqrt{1513}}{2*3}=\frac{123-3\sqrt{1513}}{6}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-123)+3\sqrt{1513}}{2*3}=\frac{123+3\sqrt{1513}}{6}
$