1/3^y=27^y+1

Solution for 1/3^y=27^y+1 equation:

1/3^y=27^y+1

We move all terms to the left:

1/3^y-(27^y+1)=0


Domain of the equation: 3^y!=0

y!=0/1

y!=0

y∈R


We get rid of parentheses

1/3^y-27^y-1=0

We multiply all the terms by the denominator


-27^y*3^y-1*3^y+1=0

Wy multiply elements

-81y^2-3y+1=0

a = -81; b = -3; c = +1;
Δ = b2-4ac
Δ = -32-4·(-81)·1
Δ = 333
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{333}=\sqrt{9*37}=\sqrt{9}*\sqrt{37}=3\sqrt{37}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-3)-3\sqrt{37}}{2*-81}=\frac{3-3\sqrt{37}}{-162}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-3)+3\sqrt{37}}{2*-81}=\frac{3+3\sqrt{37}}{-162}
$