2-1/n=3n+16

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

2-1/n=3n+16

We move all terms to the left:

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


Domain of the equation: n!=0

n∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

We add all the numbers together, and all the variables

-14n-3n*n-1=0

Wy multiply elements

-3n^2-14n-1=0

a = -3; b = -14; c = -1;
Δ = b2-4ac
Δ = -142-4·(-3)·(-1)
Δ = 184
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:

$\sqrt{\Delta}=\sqrt{184}=\sqrt{4*46}=\sqrt{4}*\sqrt{46}=2\sqrt{46}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-14)-2\sqrt{46}}{2*-3}=\frac{14-2\sqrt{46}}{-6}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-14)+2\sqrt{46}}{2*-3}=\frac{14+2\sqrt{46}}{-6}
$