1/3n-18=n+36

Solution for 1/3n-18=n+36 equation:

1/3n-18=n+36

We move all terms to the left:

1/3n-18-(n+36)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We get rid of parentheses

1/3n-n-36-18=0

We multiply all the terms by the denominator


-n*3n-36*3n-18*3n+1=0

Wy multiply elements

-3n^2-108n-54n+1=0

We add all the numbers together, and all the variables

-3n^2-162n+1=0

a = -3; b = -162; c = +1;
Δ = b2-4ac
Δ = -1622-4·(-3)·1
Δ = 26256
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:

$\sqrt{\Delta}=\sqrt{26256}=\sqrt{16*1641}=\sqrt{16}*\sqrt{1641}=4\sqrt{1641}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-162)-4\sqrt{1641}}{2*-3}=\frac{162-4\sqrt{1641}}{-6}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-162)+4\sqrt{1641}}{2*-3}=\frac{162+4\sqrt{1641}}{-6}
$