1. **State the problem:** We are given the equation $\sqrt{k - x} = 58 - x$ where $k$ is a constant. The equation has exactly one real solution. We need to find the minimum possible value of $4k$. 2. **Analyze the equation:** The equation is $\sqrt{k - x} = 58 - x$. 3. **Domain considerations:** Since the left side is a square root, the radicand must be non-negative: $$k - x \geq 0 \implies x \leq k.$$ Also, the right side must be non-negative because the square root is always non-negative: $$58 - x \geq 0 \implies x \leq 58.$$ So the domain for $x$ is $x \leq \min(k, 58)$. 4. **Square both sides:** To solve the equation, square both sides: $$ (\sqrt{k - x})^2 = (58 - x)^2 \implies k - x = (58 - x)^2. $$ 5. **Rewrite the equation:** $$ k - x = (58 - x)^2 = (58 - x)(58 - x) = (58 - x)^2. $$ 6. **Bring all terms to one side:** $$ (58 - x)^2 + x - k = 0. $$ 7. **Expand the square:** $$ (58 - x)^2 = (58)^2 - 2 \cdot 58 \cdot x + x^2 = 3364 - 116x + x^2. $$ 8. **Substitute back:** $$ 3364 - 116x + x^2 + x - k = 0 \implies x^2 - 115x + (3364 - k) = 0. $$ 9. **Quadratic in $x$:** $$ x^2 - 115x + (3364 - k) = 0. $$ 10. **Condition for exactly one real solution:** The original equation has exactly one real solution. Since we squared the equation, extraneous solutions may appear, so we must check carefully. However, the problem states the original equation has exactly one real solution, so the quadratic must have exactly one solution that satisfies the original equation. 11. **Discriminant of quadratic:** $$ \Delta = (-115)^2 - 4 \cdot 1 \cdot (3364 - k) = 13225 - 4(3364 - k) = 13225 - 13456 + 4k = -231 + 4k. $$ For the quadratic to have real solutions, $\Delta \geq 0$: $$ -231 + 4k \geq 0 \implies 4k \geq 231 \implies k \geq \frac{231}{4} = 57.75. $$ 12. **Check the number of solutions:** For exactly one real solution, the quadratic must have a repeated root, so $\Delta = 0$: $$ -231 + 4k = 0 \implies 4k = 231 \implies k = 57.75. $$ 13. **Verify the solution satisfies the original equation:** At $k = 57.75$, the quadratic has one root: $$ x = \frac{115}{2} = 57.5. $$ Check the original equation: $$ \sqrt{k - x} = \sqrt{57.75 - 57.5} = \sqrt{0.25} = 0.5, $$ $$ 58 - x = 58 - 57.5 = 0.5. $$ Both sides equal, so the solution is valid. 14. **Conclusion:** The minimum possible value of $4k$ is $231$. **Final answer:** $$ \boxed{231} $$