3^-a-1=(1/243)

Solution for 3^-a-1=(1/243) equation:

3^-a-1=(1/243)

We move all terms to the left:

3^-a-1-((1/243))=0

We add all the numbers together, and all the variables

-a-1+3^-((+1/243))=0

We add all the numbers together, and all the variables

-1a-((+1/243))=0

We multiply all the terms by the denominator


-1a*243))-((+1=0

Wy multiply elements

-243a^2+1=0

a = -243; b = 0; c = +1;
Δ = b2-4ac
Δ = 02-4·(-243)·1
Δ = 972
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{972}=\sqrt{324*3}=\sqrt{324}*\sqrt{3}=18\sqrt{3}$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-18\sqrt{3}}{2*-243}=\frac{0-18\sqrt{3}}{-486}
=-\frac{18\sqrt{3}}{-486}
=-\frac{\sqrt{3}}{-27}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+18\sqrt{3}}{2*-243}=\frac{0+18\sqrt{3}}{-486}
=\frac{18\sqrt{3}}{-486}
=\frac{\sqrt{3}}{-27}
$