21/3+3/4n=4n+18

Solution for 21/3+3/4n=4n+18 equation:

21/3+3/4n=4n+18

We move all terms to the left:

21/3+3/4n-(4n+18)=0


Domain of the equation: 4n!=0

n!=0/4

n!=0

n∈R


We add all the numbers together, and all the variables

3/4n-(4n+18)+7=0

We get rid of parentheses

3/4n-4n-18+7=0

We multiply all the terms by the denominator


-4n*4n-18*4n+7*4n+3=0

Wy multiply elements

-16n^2-72n+28n+3=0

We add all the numbers together, and all the variables

-16n^2-44n+3=0

a = -16; b = -44; c = +3;
Δ = b2-4ac
Δ = -442-4·(-16)·3
Δ = 2128
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:

$\sqrt{\Delta}=\sqrt{2128}=\sqrt{16*133}=\sqrt{16}*\sqrt{133}=4\sqrt{133}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-44)-4\sqrt{133}}{2*-16}=\frac{44-4\sqrt{133}}{-32}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-44)+4\sqrt{133}}{2*-16}=\frac{44+4\sqrt{133}}{-32}
$