360/(n-1)-360(n+2)=6

Solution for 360/(n-1)-360(n+2)=6 equation:

360/(n-1)-360(n+2)=6

We move all terms to the left:

360/(n-1)-360(n+2)-(6)=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 parentheses

360/(n-1)-360n-720-6=0

We multiply all the terms by the denominator


-360n*(n-1)-720*(n-1)-6*(n-1)+360=0

We multiply parentheses

-360n^2+360n-720n-6n+720+6+360=0

We add all the numbers together, and all the variables

-360n^2-366n+1086=0

a = -360; b = -366; c = +1086;
Δ = b2-4ac
Δ = -3662-4·(-360)·1086
Δ = 1697796
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:

$\sqrt{\Delta}=\sqrt{1697796}=\sqrt{36*47161}=\sqrt{36}*\sqrt{47161}=6\sqrt{47161}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-366)-6\sqrt{47161}}{2*-360}=\frac{366-6\sqrt{47161}}{-720}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-366)+6\sqrt{47161}}{2*-360}=\frac{366+6\sqrt{47161}}{-720}
$