Example 1 Solve the equation $\sqrt[4]{10+x}+\sqrt[4]{7-x}=3$.


Solve by mean substitution
$$
\sqrt[4]{7-x}=\frac{3}{2}-d, \sqrt[4]{10+x}=\frac{3}{2}+d .
$$

Raising both sides of the two equations to the fourth power and then adding them yields
$$
16 d^{4}+216 d^{2}-55=0 \text {. }
$$

Solving for $d$ gives $d= \pm \frac{1}{2}$.
When $d=\frac{1}{2}$, $x_{1}=6$;
When $d=-\frac{1}{2}$, $x_{2}=-9$.