Roots of cubic equation -. Special type

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Can we find the roots of a cubic equation given that Thier sum is zero. Q-. $4x^3 - 3x + 1/2 =0$. I tried eliminating $3$ equation $3$ variables but ending up again with the same cubic in terms of beta.

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Substitute $$x \leftarrow \frac12 (y + \frac1y)$$ to get $$\frac{y^6+y^3+1}{2y^3}.$$ Then solve this as a quadratic equation in $y^3$.