6n+1=1/2n-7

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

6n+1=1/2n-7

We move all terms to the left:

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


Domain of the equation: 2n-7)!=0

n∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

12n^2+14n+2n-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}
$