16^x=1/64^x-2

Solution for 16^x=1/64^x-2 equation:

16^x=1/64^x-2

We move all terms to the left:

16^x-(1/64^x-2)=0


Domain of the equation: 64^x-2)!=0

x∈R


We get rid of parentheses

16^x-1/64^x+2=0

We multiply all the terms by the denominator


16^x*64^x+2*64^x-1=0

Wy multiply elements

1024x^2+128x-1=0

a = 1024; b = 128; c = -1;
Δ = b2-4ac
Δ = 1282-4·1024·(-1)
Δ = 20480
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:

$\sqrt{\Delta}=\sqrt{20480}=\sqrt{4096*5}=\sqrt{4096}*\sqrt{5}=64\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(128)-64\sqrt{5}}{2*1024}=\frac{-128-64\sqrt{5}}{2048}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(128)+64\sqrt{5}}{2*1024}=\frac{-128+64\sqrt{5}}{2048}
$