27=1/(3^2x-1)

Solution for 27=1/(3^2x-1) equation:

27=1/(3^2x-1)

We move all terms to the left:

27-(1/(3^2x-1))=0


Domain of the equation: (3^2x-1))!=0

x∈R


We multiply all the terms by the denominator


-(1+27*(3^2x-1))=0


We calculate terms in parentheses: -(1+27*(3^2x-1)), so:

1+27*(3^2x-1)

determiningTheFunctionDomain
27*(3^2x-1)+1

We multiply parentheses

81x^2-27+1

We add all the numbers together, and all the variables

81x^2-26

Back to the equation:

-(81x^2-26)


We get rid of parentheses

-81x^2+26=0

a = -81; b = 0; c = +26;
Δ = b2-4ac
Δ = 02-4·(-81)·26
Δ = 8424
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:

$\sqrt{\Delta}=\sqrt{8424}=\sqrt{324*26}=\sqrt{324}*\sqrt{26}=18\sqrt{26}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-18\sqrt{26}}{2*-81}=\frac{0-18\sqrt{26}}{-162}
=-\frac{18\sqrt{26}}{-162}
=-\frac{\sqrt{26}}{-9}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+18\sqrt{26}}{2*-81}=\frac{0+18\sqrt{26}}{-162}
=\frac{18\sqrt{26}}{-162}
=\frac{\sqrt{26}}{-9}
$