3^x+2=1/27*x

Solution for 3^x+2=1/27*x equation:

3^x+2=1/27x

We move all terms to the left:

3^x+2-(1/27x)=0


Domain of the equation: 27x)!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

3^x-(+1/27x)+2=0

We get rid of parentheses

3^x-1/27x+2=0

We multiply all the terms by the denominator


3^x*27x+2*27x-1=0

Wy multiply elements

81x^2+54x-1=0

a = 81; b = 54; c = -1;
Δ = b2-4ac
Δ = 542-4·81·(-1)
Δ = 3240
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:

$\sqrt{\Delta}=\sqrt{3240}=\sqrt{324*10}=\sqrt{324}*\sqrt{10}=18\sqrt{10}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(54)-18\sqrt{10}}{2*81}=\frac{-54-18\sqrt{10}}{162}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(54)+18\sqrt{10}}{2*81}=\frac{-54+18\sqrt{10}}{162}
$