n+1/3n=60

Solution for n+1/3n=60 equation:

n+1/3n=60

We move all terms to the left:

n+1/3n-(60)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We multiply all the terms by the denominator


n*3n-60*3n+1=0

Wy multiply elements

3n^2-180n+1=0

a = 3; b = -180; c = +1;
Δ = b2-4ac
Δ = -1802-4·3·1
Δ = 32388
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:

$\sqrt{\Delta}=\sqrt{32388}=\sqrt{4*8097}=\sqrt{4}*\sqrt{8097}=2\sqrt{8097}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-180)-2\sqrt{8097}}{2*3}=\frac{180-2\sqrt{8097}}{6}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-180)+2\sqrt{8097}}{2*3}=\frac{180+2\sqrt{8097}}{6}
$