/3(9)^x=27^x+1

Solution for /3(9)^x=27^x+1 equation:

/3(9)^x=27^x+1

We move all terms to the left:

/3(9)^x-(27^x+1)=0


Domain of the equation: 39^x!=0

x!=0/1

x!=0

x∈R


We get rid of parentheses

/39^x-27^x-1=0

We multiply all the terms by the denominator


-27^x*39^x-1*39^x+=0

We add all the numbers together, and all the variables

-27^x*39^x-1*39^x=0

Wy multiply elements

-1053x^2-39x=0

a = -1053; b = -39; c = 0;
Δ = b2-4ac
Δ = -392-4·(-1053)·0
Δ = 1521
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{1521}=39$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-39)-39}{2*-1053}=\frac{0}{-2106}
=0
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-39)+39}{2*-1053}=\frac{78}{-2106}
=-1/27
$