36^-3x+3=1/216^x+1

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

36^-3x+3=1/216^x+1

We move all terms to the left:

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


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

x∈R


We add all the numbers together, and all the variables

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

We get rid of parentheses

-3x-1/216^x-1=0

We multiply all the terms by the denominator


-3x*216^x-1*216^x-1=0

Wy multiply elements

-648x^2-216x-1=0

a = -648; b = -216; c = -1;
Δ = b2-4ac
Δ = -2162-4·(-648)·(-1)
Δ = 44064
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:

$\sqrt{\Delta}=\sqrt{44064}=\sqrt{1296*34}=\sqrt{1296}*\sqrt{34}=36\sqrt{34}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-216)-36\sqrt{34}}{2*-648}=\frac{216-36\sqrt{34}}{-1296}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-216)+36\sqrt{34}}{2*-648}=\frac{216+36\sqrt{34}}{-1296}
$