Example 7 Given that $a$ is a real number, and makes the quadratic equation in $x$, $x^{2}+a^{2} x+a=0$, have real roots. Find the maximum value that the root $x$ of the equation can take.
(1994, Beijing Junior High School Mathematics Competition).
Solution: Consider $a$ as the unknown. When $x \neq 0$, the condition for the equation $x a^{2} + a + x^{2} = 0$ to have real roots $a$ is
$$
\Delta = 1 - 4 \cdot x^{3} \geqslant 0 \Rightarrow x \leqslant \frac{3}{2} \text{.}
$$

Substituting $x = \frac{\sqrt[3]{2}}{2}$ into the original equation and completing the square, we get
$$
\left(a - \frac{\sqrt[3]{4}}{2}\right)^{2} = 0 \text{.}
$$

Therefore, when $a = \frac{\sqrt[3]{4}}{2}$, the root $x$ has the maximum value $\frac{\sqrt[3]{2}}{2}$.