3^x+3/27^x+3=9

Solution for 3^x+3/27^x+3=9 equation:

3^x+3/27^x+3=9

We move all terms to the left:

3^x+3/27^x+3-(9)=0


Domain of the equation: 27^x!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

3^x+3/27^x-6=0

We multiply all the terms by the denominator


3^x*27^x-6*27^x+3=0

Wy multiply elements

81x^2-162x+3=0

a = 81; b = -162; c = +3;
Δ = b2-4ac
Δ = -1622-4·81·3
Δ = 25272
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:

$\sqrt{\Delta}=\sqrt{25272}=\sqrt{324*78}=\sqrt{324}*\sqrt{78}=18\sqrt{78}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-162)-18\sqrt{78}}{2*81}=\frac{162-18\sqrt{78}}{162}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-162)+18\sqrt{78}}{2*81}=\frac{162+18\sqrt{78}}{162}
$