n+4/3n=14

Solution for n+4/3n=14 equation:

n+4/3n=14

We move all terms to the left:

n+4/3n-(14)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We multiply all the terms by the denominator


n*3n-14*3n+4=0

Wy multiply elements

3n^2-42n+4=0

a = 3; b = -42; c = +4;
Δ = b2-4ac
Δ = -422-4·3·4
Δ = 1716
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:

$\sqrt{\Delta}=\sqrt{1716}=\sqrt{4*429}=\sqrt{4}*\sqrt{429}=2\sqrt{429}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-42)-2\sqrt{429}}{2*3}=\frac{42-2\sqrt{429}}{6}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-42)+2\sqrt{429}}{2*3}=\frac{42+2\sqrt{429}}{6}
$