8^2x-1=1/512

Solution for 8^2x-1=1/512 equation:

8^2x-1=1/512

We move all terms to the left:

8^2x-1-(1/512)=0

We add all the numbers together, and all the variables

8^2x-1-(+1/512)=0

We get rid of parentheses

8^2x-1-1/512=0

We multiply all the terms by the denominator


8^2x*512-1-1*512=0

We add all the numbers together, and all the variables

8^2x*512-513=0

Wy multiply elements

4096x^2-513=0

a = 4096; b = 0; c = -513;
Δ = b2-4ac
Δ = 02-4·4096·(-513)
Δ = 8404992
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:

$\sqrt{\Delta}=\sqrt{8404992}=\sqrt{147456*57}=\sqrt{147456}*\sqrt{57}=384\sqrt{57}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-384\sqrt{57}}{2*4096}=\frac{0-384\sqrt{57}}{8192}
=-\frac{384\sqrt{57}}{8192}
=-\frac{3\sqrt{57}}{64}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+384\sqrt{57}}{2*4096}=\frac{0+384\sqrt{57}}{8192}
=\frac{384\sqrt{57}}{8192}
=\frac{3\sqrt{57}}{64}
$