3^2x-1=1/243

Solution for 3^2x-1=1/243 equation:

3^2x-1=1/243

We move all terms to the left:

3^2x-1-(1/243)=0

We add all the numbers together, and all the variables

3^2x-1-(+1/243)=0

We get rid of parentheses

3^2x-1-1/243=0

We multiply all the terms by the denominator


3^2x*243-1-1*243=0

We add all the numbers together, and all the variables

3^2x*243-244=0

Wy multiply elements

729x^2-244=0

a = 729; b = 0; c = -244;
Δ = b2-4ac
Δ = 02-4·729·(-244)
Δ = 711504
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:

$\sqrt{\Delta}=\sqrt{711504}=\sqrt{11664*61}=\sqrt{11664}*\sqrt{61}=108\sqrt{61}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-108\sqrt{61}}{2*729}=\frac{0-108\sqrt{61}}{1458}
=-\frac{108\sqrt{61}}{1458}
=-\frac{2\sqrt{61}}{27}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+108\sqrt{61}}{2*729}=\frac{0+108\sqrt{61}}{1458}
=\frac{108\sqrt{61}}{1458}
=\frac{2\sqrt{61}}{27}
$