1/3x=3(2)^x

Solution for 1/3x=3(2)^x equation:

1/3x=3(2)^x

We move all terms to the left:

1/3x-(3(2)^x)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


determiningTheFunctionDomain
1/3x-32^x=0

We multiply all the terms by the denominator


-32^x*3x+1=0

Wy multiply elements

-96x^2+1=0

a = -96; b = 0; c = +1;
Δ = b2-4ac
Δ = 02-4·(-96)·1
Δ = 384
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{384}=\sqrt{64*6}=\sqrt{64}*\sqrt{6}=8\sqrt{6}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-8\sqrt{6}}{2*-96}=\frac{0-8\sqrt{6}}{-192}
=-\frac{8\sqrt{6}}{-192}
=-\frac{\sqrt{6}}{-24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+8\sqrt{6}}{2*-96}=\frac{0+8\sqrt{6}}{-192}
=\frac{8\sqrt{6}}{-192}
=\frac{\sqrt{6}}{-24}
$