1/2n+7-2n-14=5n+1

Solution for 1/2n+7-2n-14=5n+1 equation:

1/2n+7-2n-14=5n+1

We move all terms to the left:

1/2n+7-2n-14-(5n+1)=0


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We add all the numbers together, and all the variables

-2n+1/2n-(5n+1)-7=0

We get rid of parentheses

-2n+1/2n-5n-1-7=0

We multiply all the terms by the denominator


-2n*2n-5n*2n-1*2n-7*2n+1=0

Wy multiply elements

-4n^2-10n^2-2n-14n+1=0

We add all the numbers together, and all the variables

-14n^2-16n+1=0

a = -14; b = -16; c = +1;
Δ = b2-4ac
Δ = -162-4·(-14)·1
Δ = 312
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:

$\sqrt{\Delta}=\sqrt{312}=\sqrt{4*78}=\sqrt{4}*\sqrt{78}=2\sqrt{78}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-16)-2\sqrt{78}}{2*-14}=\frac{16-2\sqrt{78}}{-28}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-16)+2\sqrt{78}}{2*-14}=\frac{16+2\sqrt{78}}{-28}
$