n+2/3n=-30

Solution for n+2/3n=-30 equation:

n+2/3n=-30

We move all terms to the left:

n+2/3n-(-30)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We add all the numbers together, and all the variables

n+2/3n+30=0

We multiply all the terms by the denominator


n*3n+30*3n+2=0

Wy multiply elements

3n^2+90n+2=0

a = 3; b = 90; c = +2;
Δ = b2-4ac
Δ = 902-4·3·2
Δ = 8076
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:

$\sqrt{\Delta}=\sqrt{8076}=\sqrt{4*2019}=\sqrt{4}*\sqrt{2019}=2\sqrt{2019}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(90)-2\sqrt{2019}}{2*3}=\frac{-90-2\sqrt{2019}}{6}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(90)+2\sqrt{2019}}{2*3}=\frac{-90+2\sqrt{2019}}{6}
$