1/2n-7=6n+1

Solution for 1/2n-7=6n+1 equation:

1/2n-7=6n+1

We move all terms to the left:

1/2n-7-(6n+1)=0


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We get rid of parentheses

1/2n-6n-1-7=0

We multiply all the terms by the denominator


-6n*2n-1*2n-7*2n+1=0

Wy multiply elements

-12n^2-2n-14n+1=0

We add all the numbers together, and all the variables

-12n^2-16n+1=0

a = -12; b = -16; c = +1;
Δ = b2-4ac
Δ = -162-4·(-12)·1
Δ = 304
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:

$\sqrt{\Delta}=\sqrt{304}=\sqrt{16*19}=\sqrt{16}*\sqrt{19}=4\sqrt{19}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-16)-4\sqrt{19}}{2*-12}=\frac{16-4\sqrt{19}}{-24}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-16)+4\sqrt{19}}{2*-12}=\frac{16+4\sqrt{19}}{-24}
$