(81/27^-x)=3^x^2

Solution for (81/27^-x)=3^x^2 equation:

(81/27^-x)=3^x^2

We move all terms to the left:

(81/27^-x)-(3^x^2)=0


Domain of the equation: 27^-x)!=0

x∈R


We get rid of parentheses

-x-3^x^2+81/27^=0

We multiply all the terms by the denominator


-x*27^-3^x^2*27^+81=0

Wy multiply elements

-27x^2-81x^2+81=0

We add all the numbers together, and all the variables

-108x^2+81=0

a = -108; b = 0; c = +81;
Δ = b2-4ac
Δ = 02-4·(-108)·81
Δ = 34992
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:

$\sqrt{\Delta}=\sqrt{34992}=\sqrt{11664*3}=\sqrt{11664}*\sqrt{3}=108\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-108\sqrt{3}}{2*-108}=\frac{0-108\sqrt{3}}{-216}
=-\frac{108\sqrt{3}}{-216}
=-\frac{\sqrt{3}}{-2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+108\sqrt{3}}{2*-108}=\frac{0+108\sqrt{3}}{-216}
=\frac{108\sqrt{3}}{-216}
=\frac{\sqrt{3}}{-2}
$