3n(n+4)=8-1/2

Solution for 3n(n+4)=8-1/2 equation:

3n(n+4)=8-1/2

We move all terms to the left:

3n(n+4)-(8-1/2)=0

We add all the numbers together, and all the variables

3n(n+4)-(-1/2+8)=0

We multiply parentheses

3n^2+12n-(-1/2+8)=0

We get rid of parentheses

3n^2+12n-8+1/2=0

We multiply all the terms by the denominator


3n^2*2+12n*2+1-8*2=0

We add all the numbers together, and all the variables

3n^2*2+12n*2-15=0

Wy multiply elements

6n^2+24n-15=0

a = 6; b = 24; c = -15;
Δ = b2-4ac
Δ = 242-4·6·(-15)
Δ = 936
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:

$\sqrt{\Delta}=\sqrt{936}=\sqrt{36*26}=\sqrt{36}*\sqrt{26}=6\sqrt{26}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(24)-6\sqrt{26}}{2*6}=\frac{-24-6\sqrt{26}}{12}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(24)+6\sqrt{26}}{2*6}=\frac{-24+6\sqrt{26}}{12}
$