2/3n+n=10

Solution for 2/3n+n=10 equation:

2/3n+n=10

We move all terms to the left:

2/3n+n-(10)=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-10=0

We multiply all the terms by the denominator


n*3n-10*3n+2=0

Wy multiply elements

3n^2-30n+2=0

a = 3; b = -30; c = +2;
Δ = b2-4ac
Δ = -302-4·3·2
Δ = 876
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:

$\sqrt{\Delta}=\sqrt{876}=\sqrt{4*219}=\sqrt{4}*\sqrt{219}=2\sqrt{219}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-30)-2\sqrt{219}}{2*3}=\frac{30-2\sqrt{219}}{6}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-30)+2\sqrt{219}}{2*3}=\frac{30+2\sqrt{219}}{6}
$