2-(1)/(2)n=3n+16

Solution for 2-(1)/(2)n=3n+16 equation:

2-(1)/(2)n=3n+16

We move all terms to the left:

2-(1)/(2)n-(3n+16)=0


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We get rid of parentheses

-1/2n-3n-16+2=0

We multiply all the terms by the denominator


-3n*2n-16*2n+2*2n-1=0

Wy multiply elements

-6n^2-32n+4n-1=0

We add all the numbers together, and all the variables

-6n^2-28n-1=0

a = -6; b = -28; c = -1;
Δ = b2-4ac
Δ = -282-4·(-6)·(-1)
Δ = 760
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:

$\sqrt{\Delta}=\sqrt{760}=\sqrt{4*190}=\sqrt{4}*\sqrt{190}=2\sqrt{190}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-28)-2\sqrt{190}}{2*-6}=\frac{28-2\sqrt{190}}{-12}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-28)+2\sqrt{190}}{2*-6}=\frac{28+2\sqrt{190}}{-12}
$