[ higher degree equations (other).]

Solve the equation $x^{3}-x-\frac{2}{3 \sqrt{3}}=0$. How many real roots does it have?

#
Cardano's formula gives $x_{1}=\sqrt[3]{\frac{1}{3 \sqrt{3}}+\sqrt{\frac{1}{27}-\frac{1}{27}}}+\sqrt[3]{\frac{1}{3 \sqrt{3}}-\sqrt{\frac{1}{27}-\frac{1}{27}}}=\frac{2}{\sqrt{3}}$. Since

the expression under the square root in Cardano's formula is zero, the equation must have another multiple root (see

problem $\underline{61265)}$. Indeed, dividing by $x-x_{1}$, we obtain the quadratic equation

$x^{2}+x_{1} x+1 / 4=0$, from which $x_{2,3}=-1 / 2 x_{1}$.

## Answer

$x_{1}=\frac{2}{\sqrt{3}}, x_{2,3}=-\frac{1}{\sqrt{3}}$