n+1/5n=24

Solution for n+1/5n=24 equation:

n+1/5n=24

We move all terms to the left:

n+1/5n-(24)=0


Domain of the equation: 5n!=0

n!=0/5

n!=0

n∈R


We multiply all the terms by the denominator


n*5n-24*5n+1=0

Wy multiply elements

5n^2-120n+1=0

a = 5; b = -120; c = +1;
Δ = b2-4ac
Δ = -1202-4·5·1
Δ = 14380
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:

$\sqrt{\Delta}=\sqrt{14380}=\sqrt{4*3595}=\sqrt{4}*\sqrt{3595}=2\sqrt{3595}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-120)-2\sqrt{3595}}{2*5}=\frac{120-2\sqrt{3595}}{10}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-120)+2\sqrt{3595}}{2*5}=\frac{120+2\sqrt{3595}}{10}
$