3^x^2+2=9.1/27^x

Solution for 3^x^2+2=9.1/27^x equation:

3^x^2+2=9.1/27^x

We move all terms to the left:

3^x^2+2-(9.1/27^x)=0


Domain of the equation: 27^x)!=0

x!=0/1

x!=0

x∈R


We get rid of parentheses

3^x^2-9.1/27^x+2=0

We multiply all the terms by the denominator


3^x^2*27^x+2*27^x-9.1=0

Wy multiply elements

81x^2+54x-9.1=0

a = 81; b = 54; c = -9.1;
Δ = b2-4ac
Δ = 542-4·81·(-9.1)
Δ = 5864.4
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(54)-\sqrt{5864.4}}{2*81}=\frac{-54-\sqrt{5864.4}}{162}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(54)+\sqrt{5864.4}}{2*81}=\frac{-54+\sqrt{5864.4}}{162}
$