16^-3x=(1/64)^x+3

Solution for 16^-3x=(1/64)^x+3 equation:

16^-3x=(1/64)^x+3

We move all terms to the left:

16^-3x-((1/64)^x+3)=0


Domain of the equation: 64)^x+3)!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

-3x-((+1/64)^x+3)+16^=0

We add all the numbers together, and all the variables

-3x-((+1/64)^x+3)=0

We multiply all the terms by the denominator


-3x*64)^x+3)-((+1=0

Wy multiply elements

-192x^2+1=0

a = -192; b = 0; c = +1;
Δ = b2-4ac
Δ = 02-4·(-192)·1
Δ = 768
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:

$\sqrt{\Delta}=\sqrt{768}=\sqrt{256*3}=\sqrt{256}*\sqrt{3}=16\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-16\sqrt{3}}{2*-192}=\frac{0-16\sqrt{3}}{-384}
=-\frac{16\sqrt{3}}{-384}
=-\frac{\sqrt{3}}{-24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+16\sqrt{3}}{2*-192}=\frac{0+16\sqrt{3}}{-384}
=\frac{16\sqrt{3}}{-384}
=\frac{\sqrt{3}}{-24}
$