2-2/1n=3n+16

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

2-2/1n=3n+16

We move all terms to the left:

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


Domain of the equation: 1n!=0

n∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

-3n^2-16n+2n-2=0

We add all the numbers together, and all the variables

-3n^2-14n-2=0

a = -3; b = -14; c = -2;
Δ = b2-4ac
Δ = -142-4·(-3)·(-2)
Δ = 172
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:

$\sqrt{\Delta}=\sqrt{172}=\sqrt{4*43}=\sqrt{4}*\sqrt{43}=2\sqrt{43}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-14)-2\sqrt{43}}{2*-3}=\frac{14-2\sqrt{43}}{-6}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-14)+2\sqrt{43}}{2*-3}=\frac{14+2\sqrt{43}}{-6}
$