1/3n-10=2n

Solution for 1/3n-10=2n equation:

1/3n-10=2n

We move all terms to the left:

1/3n-10-(2n)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We add all the numbers together, and all the variables

-2n+1/3n-10=0

We multiply all the terms by the denominator


-2n*3n-10*3n+1=0

Wy multiply elements

-6n^2-30n+1=0

a = -6; b = -30; c = +1;
Δ = b2-4ac
Δ = -302-4·(-6)·1
Δ = 924
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:

$\sqrt{\Delta}=\sqrt{924}=\sqrt{4*231}=\sqrt{4}*\sqrt{231}=2\sqrt{231}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-30)-2\sqrt{231}}{2*-6}=\frac{30-2\sqrt{231}}{-12}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-30)+2\sqrt{231}}{2*-6}=\frac{30+2\sqrt{231}}{-12}
$