(3/2)(16n+3)=24n

Solution for (3/2)(16n+3)=24n equation:

(3/2)(16n+3)=24n

We move all terms to the left:

(3/2)(16n+3)-(24n)=0


Domain of the equation: 2)(16n+3)!=0

n∈R


We add all the numbers together, and all the variables

(+3/2)(16n+3)-24n=0

We add all the numbers together, and all the variables

-24n+(+3/2)(16n+3)=0

We multiply parentheses ..

(+48n^2+3/2*3)-24n=0

We multiply all the terms by the denominator


(+48n^2+3-24n*2*3)=0

We get rid of parentheses

48n^2-24n*2*3+3=0

Wy multiply elements

48n^2-144n*3+3=0

Wy multiply elements

48n^2-432n+3=0

a = 48; b = -432; c = +3;
Δ = b2-4ac
Δ = -4322-4·48·3
Δ = 186048
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:

$\sqrt{\Delta}=\sqrt{186048}=\sqrt{576*323}=\sqrt{576}*\sqrt{323}=24\sqrt{323}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-432)-24\sqrt{323}}{2*48}=\frac{432-24\sqrt{323}}{96}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-432)+24\sqrt{323}}{2*48}=\frac{432+24\sqrt{323}}{96}
$