3(n+4)=(1/2)(6n+4)

Solution for 3(n+4)=(1/2)(6n+4) equation:

3(n+4)=(1/2)(6n+4)

We move all terms to the left:

3(n+4)-((1/2)(6n+4))=0


Domain of the equation: 2)(6n+4))!=0

n∈R


We add all the numbers together, and all the variables

3(n+4)-((+1/2)(6n+4))=0

We multiply parentheses

3n-((+1/2)(6n+4))+12=0

We multiply parentheses ..

-((+6n^2+1/2*4))+3n+12=0

We multiply all the terms by the denominator


-((+6n^2+1+3n*2*4))+12*2*4))=0


We calculate terms in parentheses: -((+6n^2+1+3n*2*4)), so:

(+6n^2+1+3n*2*4)

We get rid of parentheses

6n^2+3n*2*4+1

Wy multiply elements

6n^2+24n*4+1

Wy multiply elements

6n^2+96n+1

Back to the equation:

-(6n^2+96n+1)


We add all the numbers together, and all the variables

-(6n^2+96n+1)=0

We get rid of parentheses

-6n^2-96n-1=0

a = -6; b = -96; c = -1;
Δ = b2-4ac
Δ = -962-4·(-6)·(-1)
Δ = 9192
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:

$\sqrt{\Delta}=\sqrt{9192}=\sqrt{4*2298}=\sqrt{4}*\sqrt{2298}=2\sqrt{2298}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-96)-2\sqrt{2298}}{2*-6}=\frac{96-2\sqrt{2298}}{-12}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-96)+2\sqrt{2298}}{2*-6}=\frac{96+2\sqrt{2298}}{-12}
$