n+1/3n=24

Solution for n+1/3n=24 equation:

n+1/3n=24

We move all terms to the left:

n+1/3n-(24)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We multiply all the terms by the denominator


n*3n-24*3n+1=0

Wy multiply elements

3n^2-72n+1=0

a = 3; b = -72; c = +1;
Δ = b2-4ac
Δ = -722-4·3·1
Δ = 5172
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:

$\sqrt{\Delta}=\sqrt{5172}=\sqrt{4*1293}=\sqrt{4}*\sqrt{1293}=2\sqrt{1293}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-72)-2\sqrt{1293}}{2*3}=\frac{72-2\sqrt{1293}}{6}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-72)+2\sqrt{1293}}{2*3}=\frac{72+2\sqrt{1293}}{6}
$