52=n-1/5n

Solution for 52=n-1/5n equation:

52=n-1/5n

We move all terms to the left:

52-(n-1/5n)=0


Domain of the equation: 5n)!=0

n!=0/1

n!=0

n∈R


We add all the numbers together, and all the variables

-(+n-1/5n)+52=0

We get rid of parentheses

-n+1/5n+52=0

We multiply all the terms by the denominator


-n*5n+52*5n+1=0

Wy multiply elements

-5n^2+260n+1=0

a = -5; b = 260; c = +1;
Δ = b2-4ac
Δ = 2602-4·(-5)·1
Δ = 67620
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:

$\sqrt{\Delta}=\sqrt{67620}=\sqrt{196*345}=\sqrt{196}*\sqrt{345}=14\sqrt{345}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(260)-14\sqrt{345}}{2*-5}=\frac{-260-14\sqrt{345}}{-10}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(260)+14\sqrt{345}}{2*-5}=\frac{-260+14\sqrt{345}}{-10}
$