9^-x=3^-x/3*81^x

Solution for 9^-x=3^-x/3*81^x equation:

9^-x=3^-x/3*81^x

We move all terms to the left:

9^-x-(3^-x/3*81^x)=0


Domain of the equation: 3*81^x)!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

-1x-(3^-x/3*81^x)=0

We get rid of parentheses

-1x+x/3*81^x-3^=0

We multiply all the terms by the denominator


-1x*3*81^x+x-3^*3*81^x=0

We add all the numbers together, and all the variables

x-1x*3*81^x-3^*3*81^x=0

Wy multiply elements

-243x^2*8-729x^2*8+x=0

Wy multiply elements

-1944x^2-5832x^2+x=0

We add all the numbers together, and all the variables

-7776x^2+x=0

a = -7776; b = 1; c = 0;
Δ = b2-4ac
Δ = 12-4·(-7776)·0
Δ = 1
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{1}=1$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-1}{2*-7776}=\frac{-2}{-15552}
=1/7776
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+1}{2*-7776}=\frac{0}{-15552}
=0
$