7. If the equation $\left(x^{2}-1\right)\left(x^{2}-4\right)=k$ has four non-zero real roots, and the four points corresponding to them on the number line are equally spaced, find the value of $k$.
7. From $\left(x^{2}-1\right)\left(x^{2}-4\right)=k$ having four non-zero real roots $x_{1}, x_{2}, x_{3}, x_{4}\left(x_{1}<x_{2}<x_{3}<x_{4}\right)$, we know that the equation
$$
(t-1)(t-4)=k \Rightarrow t^{2}-5 t+4-k=0
$$

has two distinct positive real roots, and $x_{1}, x_{2}, x_{3}, x_{4}$ correspond to four points on the number line that are equally spaced, which must be symmetrically arranged around 0.
Let $x_{3}=m$. Then
$$
x_{3}-x_{2}=2 m, x_{4}=x_{3}+2 m=3 m \text {. }
$$

If $t_{1}=m^{2}$, then $t_{2}=9 m^{2} \Rightarrow t_{2}=9 t_{1}$.
By Vieta's formulas, we have $\left\{\begin{array}{l}t_{1}+9 t_{1}=5, \\ t_{1} \cdot 9 t_{1}=4-k,\end{array}\right.$ which gives $k=\frac{7}{4}$.