12^x*3^x+1=648

Solution for 12^x*3^x+1=648 equation:

12^x*3^x+1=648

We move all terms to the left:

12^x*3^x+1-(648)=0

We add all the numbers together, and all the variables

12^x*3^x-647=0

Wy multiply elements

36x^2-647=0

a = 36; b = 0; c = -647;
Δ = b2-4ac
Δ = 02-4·36·(-647)
Δ = 93168
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:

$\sqrt{\Delta}=\sqrt{93168}=\sqrt{144*647}=\sqrt{144}*\sqrt{647}=12\sqrt{647}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-12\sqrt{647}}{2*36}=\frac{0-12\sqrt{647}}{72}
=-\frac{12\sqrt{647}}{72}
=-\frac{\sqrt{647}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+12\sqrt{647}}{2*36}=\frac{0+12\sqrt{647}}{72}
=\frac{12\sqrt{647}}{72}
=\frac{\sqrt{647}}{6}
$