81^x=9/3^x

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

81^x=9/3^x

We move all terms to the left:

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


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

x!=0/1

x!=0

x∈R


We get rid of parentheses

81^x-9/3^x=0

We multiply all the terms by the denominator


81^x*3^x-9=0

Wy multiply elements

243x^2-9=0

a = 243; b = 0; c = -9;
Δ = b2-4ac
Δ = 02-4·243·(-9)
Δ = 8748
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:

$\sqrt{\Delta}=\sqrt{8748}=\sqrt{2916*3}=\sqrt{2916}*\sqrt{3}=54\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-54\sqrt{3}}{2*243}=\frac{0-54\sqrt{3}}{486}
=-\frac{54\sqrt{3}}{486}
=-\frac{\sqrt{3}}{9}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+54\sqrt{3}}{2*243}=\frac{0+54\sqrt{3}}{486}
=\frac{54\sqrt{3}}{486}
=\frac{\sqrt{3}}{9}
$