36^-2x+3=(1/216)^x+1

Solution for 36^-2x+3=(1/216)^x+1 equation:

36^-2x+3=(1/216)^x+1

We move all terms to the left:

36^-2x+3-((1/216)^x+1)=0


Domain of the equation: 216)^x+1)!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

-2x-((+1/216)^x+1)+3+36^=0

We add all the numbers together, and all the variables

-2x-((+1/216)^x+1)=0

We multiply all the terms by the denominator


-2x*216)^x+1)-((+1=0

Wy multiply elements

-432x^2+1=0

a = -432; b = 0; c = +1;
Δ = b2-4ac
Δ = 02-4·(-432)·1
Δ = 1728
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:

$\sqrt{\Delta}=\sqrt{1728}=\sqrt{576*3}=\sqrt{576}*\sqrt{3}=24\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-24\sqrt{3}}{2*-432}=\frac{0-24\sqrt{3}}{-864}
=-\frac{24\sqrt{3}}{-864}
=-\frac{\sqrt{3}}{-36}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+24\sqrt{3}}{2*-432}=\frac{0+24\sqrt{3}}{-864}
=\frac{24\sqrt{3}}{-864}
=\frac{\sqrt{3}}{-36}
$