3^-2x=(1/81)

Solution for 3^-2x=(1/81) equation:

3^-2x=(1/81)

We move all terms to the left:

3^-2x-((1/81))=0

We add all the numbers together, and all the variables

-2x+3^-((+1/81))=0

We add all the numbers together, and all the variables

-2x-((+1/81))=0

We multiply all the terms by the denominator


-2x*81))-((+1=0

Wy multiply elements

-162x^2+1=0

a = -162; b = 0; c = +1;
Δ = b2-4ac
Δ = 02-4·(-162)·1
Δ = 648
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:

$\sqrt{\Delta}=\sqrt{648}=\sqrt{324*2}=\sqrt{324}*\sqrt{2}=18\sqrt{2}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-18\sqrt{2}}{2*-162}=\frac{0-18\sqrt{2}}{-324}
=-\frac{18\sqrt{2}}{-324}
=-\frac{\sqrt{2}}{-18}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+18\sqrt{2}}{2*-162}=\frac{0+18\sqrt{2}}{-324}
=\frac{18\sqrt{2}}{-324}
=\frac{\sqrt{2}}{-18}
$