((n-2)180)/n=140+n

Solution for ((n-2)180)/n=140+n equation:

((n-2)180)/n=140+n

We move all terms to the left:

((n-2)180)/n-(140+n)=0


Domain of the equation: n!=0

n∈R


We add all the numbers together, and all the variables

((n-2)180)/n-(n+140)=0

We get rid of parentheses

((n-2)180)/n-n-140=0

We multiply all the terms by the denominator


((n-2)180)-n*n-140*n=0


We calculate terms in parentheses: +((n-2)180), so:

(n-2)180

We multiply parentheses

180n-360

Back to the equation:

+(180n-360)


We add all the numbers together, and all the variables

-140n+(180n-360)-n*n=0

Wy multiply elements

-1n^2-140n+(180n-360)=0

We get rid of parentheses

-1n^2-140n+180n-360=0

We add all the numbers together, and all the variables

-1n^2+40n-360=0

a = -1; b = 40; c = -360;
Δ = b2-4ac
Δ = 402-4·(-1)·(-360)
Δ = 160
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:

$\sqrt{\Delta}=\sqrt{160}=\sqrt{16*10}=\sqrt{16}*\sqrt{10}=4\sqrt{10}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(40)-4\sqrt{10}}{2*-1}=\frac{-40-4\sqrt{10}}{-2}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(40)+4\sqrt{10}}{2*-1}=\frac{-40+4\sqrt{10}}{-2}
$