1/3n+n=36

Solution for 1/3n+n=36 equation:

1/3n+n=36

We move all terms to the left:

1/3n+n-(36)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We add all the numbers together, and all the variables

n+1/3n-36=0

We multiply all the terms by the denominator


n*3n-36*3n+1=0

Wy multiply elements

3n^2-108n+1=0

a = 3; b = -108; c = +1;
Δ = b2-4ac
Δ = -1082-4·3·1
Δ = 11652
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:

$\sqrt{\Delta}=\sqrt{11652}=\sqrt{4*2913}=\sqrt{4}*\sqrt{2913}=2\sqrt{2913}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-108)-2\sqrt{2913}}{2*3}=\frac{108-2\sqrt{2913}}{6}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-108)+2\sqrt{2913}}{2*3}=\frac{108+2\sqrt{2913}}{6}
$