To solve this equation, we can try factoring by grouping:

8{x}^{3} - 8{x}^{2} + 1 = 0

First, we can factor out the common factor of 8{x}^{2} from the first two terms:

8{x}^{2}(x - 1) + 1 = 0

Now we have a quadratic expression in terms of (x - 1) that we can factor further:

Let y = x - 1

8{x}^{2}y + 1 = 0

This is in the form of a quadratic equation, and we can solve for y:

8{x}^{2}y = -1
y = -\frac{1}{8{x}^{2}}

Now, substitute back y = x - 1:

x - 1 = -\frac{1}{8x^2}

Solving for x by cross multiplying:

8x^{3} + 8x^{2} - 1 = 0

This is a cubic equation that can be solved using various methods like factoring, cubic formula, or numerical methods.