Example 9 If the equation $\left(x^{2}-1\right)\left(x^{2}-4\right)=k$ has 4 non-zero real roots, and the 4 points corresponding to them on the number line are equally spaced, then $k=$ $\qquad$
Solution: Since the given equation is a biquadratic equation, its 4 non-zero real roots correspond to 4 equidistant points on the number line that are symmetric about the origin. Therefore, we can set
$$
\begin{array}{l}
\left(x^{2}-1\right)\left(x^{2}-4\right)-k \\
=(x+3 a)(x+a)(x-a)(x-3 a),
\end{array}
$$

which is
$$
\begin{array}{l}
x^{4}-5 x^{2}+4-k \\
=x^{4}-10 a^{2} x^{2}+9 a^{4} .
\end{array}
$$

By comparing the coefficients of like terms on both sides of the equation, we get
$$
\left\{\begin{array}{l}
10 a^{2}=5, \\
4-k=9 a^{4} .
\end{array}\right.
$$

From this, we can obtain $\bar{k}=\frac{7}{4}$.