2/3n=90n

Solution for 2/3n=90n equation:

2/3n=90n

We move all terms to the left:

2/3n-(90n)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We add all the numbers together, and all the variables

-90n+2/3n=0

We multiply all the terms by the denominator


-90n*3n+2=0

Wy multiply elements

-270n^2+2=0

a = -270; b = 0; c = +2;
Δ = b2-4ac
Δ = 02-4·(-270)·2
Δ = 2160
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:

$\sqrt{\Delta}=\sqrt{2160}=\sqrt{144*15}=\sqrt{144}*\sqrt{15}=12\sqrt{15}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-12\sqrt{15}}{2*-270}=\frac{0-12\sqrt{15}}{-540}
=-\frac{12\sqrt{15}}{-540}
=-\frac{\sqrt{15}}{-45}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+12\sqrt{15}}{2*-270}=\frac{0+12\sqrt{15}}{-540}
=\frac{12\sqrt{15}}{-540}
=\frac{\sqrt{15}}{-45}
$