1/3n+n=8

Solution for 1/3n+n=8 equation:

1/3n+n=8

We move all terms to the left:

1/3n+n-(8)=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-8=0

We multiply all the terms by the denominator


n*3n-8*3n+1=0

Wy multiply elements

3n^2-24n+1=0

a = 3; b = -24; c = +1;
Δ = b2-4ac
Δ = -242-4·3·1
Δ = 564
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:

$\sqrt{\Delta}=\sqrt{564}=\sqrt{4*141}=\sqrt{4}*\sqrt{141}=2\sqrt{141}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-24)-2\sqrt{141}}{2*3}=\frac{24-2\sqrt{141}}{6}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-24)+2\sqrt{141}}{2*3}=\frac{24+2\sqrt{141}}{6}
$