1. **State the problem:** We need to make $d$ the subject of the formula given by $$P = \frac{1}{2} mn^2 - qd^2.$$ 2. **Rewrite the formula:** Start by isolating the term with $d^2$ on one side: $$P = \frac{1}{2} mn^2 - qd^2 \implies qd^2 = \frac{1}{2} mn^2 - P.$$ 3. **Solve for $d^2$:** Divide both sides by $q$: $$d^2 = \frac{\frac{1}{2} mn^2 - P}{q} = \frac{1}{2q} mn^2 - \frac{P}{q}.$$ 4. **Take the square root:** To solve for $d$, take the square root of both sides, remembering to consider both positive and negative roots: $$d = \pm \sqrt{\frac{1}{2q} mn^2 - \frac{P}{q}}.$$ 5. **Final formula:** $$\boxed{d = \pm \sqrt{\frac{mn^2}{2q} - \frac{P}{q}}}.$$ This expresses $d$ in terms of $P$, $m$, $n$, and $q$.