2-1/2n=3n+13

Solution for 2-1/2n=3n+13 equation:

2-1/2n=3n+13

We move all terms to the left:

2-1/2n-(3n+13)=0


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We get rid of parentheses

-1/2n-3n-13+2=0

We multiply all the terms by the denominator


-3n*2n-13*2n+2*2n-1=0

Wy multiply elements

-6n^2-26n+4n-1=0

We add all the numbers together, and all the variables

-6n^2-22n-1=0

a = -6; b = -22; c = -1;
Δ = b2-4ac
Δ = -222-4·(-6)·(-1)
Δ = 460
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:

$\sqrt{\Delta}=\sqrt{460}=\sqrt{4*115}=\sqrt{4}*\sqrt{115}=2\sqrt{115}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-22)-2\sqrt{115}}{2*-6}=\frac{22-2\sqrt{115}}{-12}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-22)+2\sqrt{115}}{2*-6}=\frac{22+2\sqrt{115}}{-12}
$