n+1/5n=30

Solution for n+1/5n=30 equation:

n+1/5n=30

We move all terms to the left:

n+1/5n-(30)=0


Domain of the equation: 5n!=0

n!=0/5

n!=0

n∈R


We multiply all the terms by the denominator


n*5n-30*5n+1=0

Wy multiply elements

5n^2-150n+1=0

a = 5; b = -150; c = +1;
Δ = b2-4ac
Δ = -1502-4·5·1
Δ = 22480
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:

$\sqrt{\Delta}=\sqrt{22480}=\sqrt{16*1405}=\sqrt{16}*\sqrt{1405}=4\sqrt{1405}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-150)-4\sqrt{1405}}{2*5}=\frac{150-4\sqrt{1405}}{10}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-150)+4\sqrt{1405}}{2*5}=\frac{150+4\sqrt{1405}}{10}
$