24.11. Solve the equations

\[
\begin{gathered}
x^{5}-10 x^{3}\left(1-x^{2}\right)+5 x\left(1-x^{2}\right)^{2}=a \\
x^{7}-21 x^{5}\left(1-x^{2}\right)+35 x^{3}\left(1-x^{2}\right)^{2}-7 x\left(1-x^{2}\right)^{3}=a
\end{gathered}
\]
24.11. Let $x=\cos \varphi$. From de Moivre's formula, it follows that the equations in question have the form $\cos 5 \varphi=a$ and $\cos 7 \varphi=a$. Therefore, the solutions to these equations are of the form

$$
x=\frac{1}{2}\left(\sqrt[n]{a+i \sqrt{1-a^{2}}}+\sqrt[n]{a-i \sqrt{1-a^{2}}}\right)
$$

where $n=5$ or 7.

## 294 Chapter 24. Equations Solvable in Radicals