[ Higher degree equations (other).]

Write down the equation of which the root will be the number $\alpha=\frac{1}{2}(\sqrt[3]{5 \sqrt{2}+7}-\sqrt[3]{5 \sqrt{2}-7})$. Write the number $\alpha$ without using radicals.
Let's write $\alpha$ in the form. By comparing with Cardano's formula (see problem $\underline{61262}$), we find $9 / 2=-7 / 8$ $q^{2} / 4+p^{3} / 27=25 / 32$, from which $q=-7 / 4, p=3 / 4$. Therefore, $\alpha$ is a root of the equation $x^{3}+3 / 4 x-7 / 4=0$. Its obvious root is $\alpha=1$.