3^2x-6=1/729

Solution for 3^2x-6=1/729 equation:

3^2x-6=1/729

We move all terms to the left:

3^2x-6-(1/729)=0

We add all the numbers together, and all the variables

3^2x-6-(+1/729)=0

We get rid of parentheses

3^2x-6-1/729=0

We multiply all the terms by the denominator


3^2x*729-1-6*729=0

We add all the numbers together, and all the variables

3^2x*729-4375=0

Wy multiply elements

2187x^2-4375=0

a = 2187; b = 0; c = -4375;
Δ = b2-4ac
Δ = 02-4·2187·(-4375)
Δ = 38272500
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:

$\sqrt{\Delta}=\sqrt{38272500}=\sqrt{1822500*21}=\sqrt{1822500}*\sqrt{21}=1350\sqrt{21}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-1350\sqrt{21}}{2*2187}=\frac{0-1350\sqrt{21}}{4374}
=-\frac{1350\sqrt{21}}{4374}
=-\frac{25\sqrt{21}}{81}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+1350\sqrt{21}}{2*2187}=\frac{0+1350\sqrt{21}}{4374}
=\frac{1350\sqrt{21}}{4374}
=\frac{25\sqrt{21}}{81}
$