1. **Stating the problem:** Solve the first equation given: $$(X - 2)^2 = (X + 7)^2 - 3X$$ 2. **Formula and rules:** To solve equations involving squares, we can expand both sides using the formula $$(a+b)^2 = a^2 + 2ab + b^2$$ and then simplify. 3. **Expand both sides:** Left side: $$(X - 2)^2 = X^2 - 4X + 4$$ Right side: $$(X + 7)^2 - 3X = (X^2 + 14X + 49) - 3X = X^2 + 11X + 49$$ 4. **Set the equation:** $$X^2 - 4X + 4 = X^2 + 11X + 49$$ 5. **Subtract $X^2$ from both sides:** $$-4X + 4 = 11X + 49$$ 6. **Bring all terms to one side:** $$-4X - 11X + 4 - 49 = 0$$ $$-15X - 45 = 0$$ 7. **Solve for $X$:** $$-15X = 45$$ $$X = \frac{45}{-15} = -3$$ **Final answer:** $$X = -3$$