72+(1/7n)=n

Solution for 72+(1/7n)=n equation:

72+(1/7n)=n

We move all terms to the left:

72+(1/7n)-(n)=0


Domain of the equation: 7n)!=0

n!=0/1

n!=0

n∈R


We add all the numbers together, and all the variables

(+1/7n)-n+72=0

We add all the numbers together, and all the variables

-1n+(+1/7n)+72=0

We get rid of parentheses

-1n+1/7n+72=0

We multiply all the terms by the denominator


-1n*7n+72*7n+1=0

Wy multiply elements

-7n^2+504n+1=0

a = -7; b = 504; c = +1;
Δ = b2-4ac
Δ = 5042-4·(-7)·1
Δ = 254044
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:

$\sqrt{\Delta}=\sqrt{254044}=\sqrt{4*63511}=\sqrt{4}*\sqrt{63511}=2\sqrt{63511}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(504)-2\sqrt{63511}}{2*-7}=\frac{-504-2\sqrt{63511}}{-14}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(504)+2\sqrt{63511}}{2*-7}=\frac{-504+2\sqrt{63511}}{-14}
$