n=12/n+8

Solution for n=12/n+8 equation:

n=12/n+8

We move all terms to the left:

n-(12/n+8)=0


Domain of the equation: n+8)!=0

n∈R


We get rid of parentheses

n-12/n-8=0

We multiply all the terms by the denominator


n*n-8*n-12=0

We add all the numbers together, and all the variables

-8n+n*n-12=0

Wy multiply elements

n^2-8n-12=0

a = 1; b = -8; c = -12;
Δ = b2-4ac
Δ = -82-4·1·(-12)
Δ = 112
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:

$\sqrt{\Delta}=\sqrt{112}=\sqrt{16*7}=\sqrt{16}*\sqrt{7}=4\sqrt{7}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-4\sqrt{7}}{2*1}=\frac{8-4\sqrt{7}}{2}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+4\sqrt{7}}{2*1}=\frac{8+4\sqrt{7}}{2}
$