2^x*3^x=364.5

Solution for 2^x*3^x=364.5 equation:

2^x*3^x=364.5

We move all terms to the left:

2^x*3^x-(364.5)=0

We add all the numbers together, and all the variables

2^x*3^x-364.5=0

Wy multiply elements

6x^2-364.5=0

a = 6; b = 0; c = -364.5;
Δ = b2-4ac
Δ = 02-4·6·(-364.5)
Δ = 8748
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:

$\sqrt{\Delta}=\sqrt{8748}=\sqrt{2916*3}=\sqrt{2916}*\sqrt{3}=54\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-54\sqrt{3}}{2*6}=\frac{0-54\sqrt{3}}{12}
=-\frac{54\sqrt{3}}{12}
=-\frac{9\sqrt{3}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+54\sqrt{3}}{2*6}=\frac{0+54\sqrt{3}}{12}
=\frac{54\sqrt{3}}{12}
=\frac{9\sqrt{3}}{2}
$