3^2x-2=1/243

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

3^2x-2=1/243

We move all terms to the left:

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

We add all the numbers together, and all the variables

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

We get rid of parentheses

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

We multiply all the terms by the denominator


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

We add all the numbers together, and all the variables

3^2x*243-487=0

Wy multiply elements

729x^2-487=0

a = 729; b = 0; c = -487;
Δ = b2-4ac
Δ = 02-4·729·(-487)
Δ = 1420092
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:

$\sqrt{\Delta}=\sqrt{1420092}=\sqrt{2916*487}=\sqrt{2916}*\sqrt{487}=54\sqrt{487}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-54\sqrt{487}}{2*729}=\frac{0-54\sqrt{487}}{1458}
=-\frac{54\sqrt{487}}{1458}
=-\frac{\sqrt{487}}{27}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+54\sqrt{487}}{2*729}=\frac{0+54\sqrt{487}}{1458}
=\frac{54\sqrt{487}}{1458}
=\frac{\sqrt{487}}{27}
$