(216)^x=(1296)^-1/4*6^x-1

Solution for (216)^x=(1296)^-1/4*6^x-1 equation:

(216)^x=(1296)^-1/4*6^x-1

We move all terms to the left:

(216)^x-((1296)^-1/4*6^x-1)=0


Domain of the equation: 4*6^x-1)!=0

x∈R


We get rid of parentheses

216^x+1/4*6^x+1-1296^=0

We multiply all the terms by the denominator


216^x*4*6^x+1*4*6^x-1296^*4*6^x+1=0

Wy multiply elements

5184x^2*6-31104x^2*6+24x*6+1=0

Wy multiply elements

31104x^2-186624x^2+144x+1=0

We add all the numbers together, and all the variables

-155520x^2+144x+1=0

a = -155520; b = 144; c = +1;
Δ = b2-4ac
Δ = 1442-4·(-155520)·1
Δ = 642816
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:

$\sqrt{\Delta}=\sqrt{642816}=\sqrt{20736*31}=\sqrt{20736}*\sqrt{31}=144\sqrt{31}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(144)-144\sqrt{31}}{2*-155520}=\frac{-144-144\sqrt{31}}{-311040}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(144)+144\sqrt{31}}{2*-155520}=\frac{-144+144\sqrt{31}}{-311040}
$