1/2n-3=4n+11

Solution for 1/2n-3=4n+11 equation:

1/2n-3=4n+11

We move all terms to the left:

1/2n-3-(4n+11)=0


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We get rid of parentheses

1/2n-4n-11-3=0

We multiply all the terms by the denominator


-4n*2n-11*2n-3*2n+1=0

Wy multiply elements

-8n^2-22n-6n+1=0

We add all the numbers together, and all the variables

-8n^2-28n+1=0

a = -8; b = -28; c = +1;
Δ = b2-4ac
Δ = -282-4·(-8)·1
Δ = 816
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:

$\sqrt{\Delta}=\sqrt{816}=\sqrt{16*51}=\sqrt{16}*\sqrt{51}=4\sqrt{51}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-28)-4\sqrt{51}}{2*-8}=\frac{28-4\sqrt{51}}{-16}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-28)+4\sqrt{51}}{2*-8}=\frac{28+4\sqrt{51}}{-16}
$