8^3-8x^2-1=0

Solution for 8^3-8x^2-1=0 equation:

8^3-8x^2-1=0

We add all the numbers together, and all the variables

-8x^2+511=0

a = -8; b = 0; c = +511;
Δ = b2-4ac
Δ = 02-4·(-8)·511
Δ = 16352
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:

$\sqrt{\Delta}=\sqrt{16352}=\sqrt{16*1022}=\sqrt{16}*\sqrt{1022}=4\sqrt{1022}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-4\sqrt{1022}}{2*-8}=\frac{0-4\sqrt{1022}}{-16}
=-\frac{4\sqrt{1022}}{-16}
=-\frac{\sqrt{1022}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+4\sqrt{1022}}{2*-8}=\frac{0+4\sqrt{1022}}{-16}
=\frac{4\sqrt{1022}}{-16}
=\frac{\sqrt{1022}}{-4}
$