3n+4=1/2n-8

Solution for 3n+4=1/2n-8 equation:

3n+4=1/2n-8

We move all terms to the left:

3n+4-(1/2n-8)=0


Domain of the equation: 2n-8)!=0

n∈R


We get rid of parentheses

3n-1/2n+8+4=0

We multiply all the terms by the denominator


3n*2n+8*2n+4*2n-1=0

Wy multiply elements

6n^2+16n+8n-1=0

We add all the numbers together, and all the variables

6n^2+24n-1=0

a = 6; b = 24; c = -1;
Δ = b2-4ac
Δ = 242-4·6·(-1)
Δ = 600
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:

$\sqrt{\Delta}=\sqrt{600}=\sqrt{100*6}=\sqrt{100}*\sqrt{6}=10\sqrt{6}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(24)-10\sqrt{6}}{2*6}=\frac{-24-10\sqrt{6}}{12}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(24)+10\sqrt{6}}{2*6}=\frac{-24+10\sqrt{6}}{12}
$