4/2^x=64^x

Solution for 4/2^x=64^x equation:

4/2^x=64^x

We move all terms to the left:

4/2^x-(64^x)=0


Domain of the equation: 2^x!=0

x!=0/1

x!=0

x∈R


We multiply all the terms by the denominator


-64^x*2^x+4=0

Wy multiply elements

-128x^2+4=0

a = -128; b = 0; c = +4;
Δ = b2-4ac
Δ = 02-4·(-128)·4
Δ = 2048
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:

$\sqrt{\Delta}=\sqrt{2048}=\sqrt{1024*2}=\sqrt{1024}*\sqrt{2}=32\sqrt{2}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-32\sqrt{2}}{2*-128}=\frac{0-32\sqrt{2}}{-256}
=-\frac{32\sqrt{2}}{-256}
=-\frac{\sqrt{2}}{-8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+32\sqrt{2}}{2*-128}=\frac{0+32\sqrt{2}}{-256}
=\frac{32\sqrt{2}}{-256}
=\frac{\sqrt{2}}{-8}
$