1/3(2n)+2n=20

Solution for 1/3(2n)+2n=20 equation:

1/3(2n)+2n=20

We move all terms to the left:

1/3(2n)+2n-(20)=0


Domain of the equation: 32n!=0

n!=0/32

n!=0

n∈R


We add all the numbers together, and all the variables

2n+1/32n-20=0

We multiply all the terms by the denominator


2n*32n-20*32n+1=0

Wy multiply elements

64n^2-640n+1=0

a = 64; b = -640; c = +1;
Δ = b2-4ac
Δ = -6402-4·64·1
Δ = 409344
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:

$\sqrt{\Delta}=\sqrt{409344}=\sqrt{256*1599}=\sqrt{256}*\sqrt{1599}=16\sqrt{1599}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-640)-16\sqrt{1599}}{2*64}=\frac{640-16\sqrt{1599}}{128}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-640)+16\sqrt{1599}}{2*64}=\frac{640+16\sqrt{1599}}{128}
$