How to find the real roots of an equation without f(x)

Question

So I have an equation of a graph, $x^4=x^2-\frac{x} {2}+\frac{1}{16}$. I need to find the number of real roots of said equation using an algebraic method. The thing is, I have no clue how to do this as there is no way I know to get $f(x)$ here. I know I can use Sturm's theorem to find the number of real roots but obviously I can't use it without $f(x)$. How can I find the number of real roots of this equation?

Answer 1

To find f(x), simply subtract $x^4$ from both sides. On the other hand, to get it in standard form let $$x^4-x^2+\frac{x}{2}-\frac{1}{16}=0$$ I think the equation can also be expressed be better for solving. Let me know if it works.

$$\frac{1}{16}(2x-1)^2(4x^2 +4x-1)=0$$