1/2*4n+18=3n+6

Solution for 1/2*4n+18=3n+6 equation:

1/2*4n+18=3n+6

We move all terms to the left:

1/2*4n+18-(3n+6)=0


Domain of the equation: 2*4n!=0

n!=0/1

n!=0

n∈R


We get rid of parentheses

1/2*4n-3n-6+18=0

We multiply all the terms by the denominator


-3n*2*4n-6*2*4n+18*2*4n+1=0

Wy multiply elements

-24n^2*4-48n*4+144n*4+1=0

Wy multiply elements

-96n^2-192n+576n+1=0

We add all the numbers together, and all the variables

-96n^2+384n+1=0

a = -96; b = 384; c = +1;
Δ = b2-4ac
Δ = 3842-4·(-96)·1
Δ = 147840
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:

$\sqrt{\Delta}=\sqrt{147840}=\sqrt{64*2310}=\sqrt{64}*\sqrt{2310}=8\sqrt{2310}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(384)-8\sqrt{2310}}{2*-96}=\frac{-384-8\sqrt{2310}}{-192}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(384)+8\sqrt{2310}}{2*-96}=\frac{-384+8\sqrt{2310}}{-192}
$