1. **State the problem:** Solve the equation $112 + (X - 1)^2 = X^2$ for $X$. 2. **Expand the squared term:** Recall that $(X - 1)^2 = X^2 - 2X + 1$. 3. Substitute into the equation: $$112 + X^2 - 2X + 1 = X^2$$ 4. **Simplify both sides:** Combine like terms on the left: $$112 + 1 + X^2 - 2X = X^2$$ $$113 + X^2 - 2X = X^2$$ 5. **Subtract $X^2$ from both sides:** $$113 + \cancel{X^2} - 2X = \cancel{X^2}$$ $$113 - 2X = 0$$ 6. **Isolate $X$:** $$-2X = -113$$ 7. **Divide both sides by $-2$:** $$X = \frac{-113}{-2} = \frac{113}{2}$$ 8. **Final answer:** $$X = \frac{113}{2} = 56.5$$