1. **State the problem:** Solve the equation $$\sqrt{x} + 4 + \sqrt{3x} + 9 = \sqrt{x} + 25$$. 2. **Simplify the equation:** Combine like terms on the left side: $$\sqrt{x} + \sqrt{3x} + 13 = \sqrt{x} + 25$$ 3. **Subtract $\sqrt{x}$ from both sides:** $$\cancel{\sqrt{x}} + \sqrt{3x} + 13 = \cancel{\sqrt{x}} + 25$$ which simplifies to $$\sqrt{3x} + 13 = 25$$ 4. **Isolate $\sqrt{3x}$:** $$\sqrt{3x} = 25 - 13$$ $$\sqrt{3x} = 12$$ 5. **Square both sides to eliminate the square root:** $$\left(\sqrt{3x}\right)^2 = 12^2$$ $$3x = 144$$ 6. **Solve for $x$:** $$x = \frac{144}{3}$$ $$x = 48$$ 7. **Check the solution:** Substitute $x=48$ back into the original equation: $$\sqrt{48} + 4 + \sqrt{3 \times 48} + 9 \stackrel{?}{=} \sqrt{48} + 25$$ Calculate each term: $$\sqrt{48} \approx 6.928$$ $$\sqrt{144} = 12$$ Left side: $$6.928 + 4 + 12 + 9 = 31.928$$ Right side: $$6.928 + 25 = 31.928$$ Both sides are equal, so $x=48$ is the solution. **Final answer:** $$x = 48$$