Question:

Step 1: Rearrange the given constraint equation to identify its structure. We start with the given condition: $$x_1^2 + \sqrt{x_2^2 - 2x_2} = 2x_1 - 1$$ Rearranging all terms to one side: $$x_1^2 - 2x_1 + 1 + \sqrt{x_2^2 - 2x_2} = 0$$ Step 2: Factor and simplify the expression. Recognize that $x_1^2 - 2x_1 + 1 = (x_1 - 1)^2$ and factor the expression under the square root: $$(x_1 - 1)^2 + \sqrt{x_2(x_2 - 2)} = 0$$ Step 3: Apply the condition that both non-negative terms must equal zero. Since we have a sum of two non-negative terms equal to zero: - $(x_1 - 1)^2 \geq 0$ (always non-negative as a perfect square) - $\sqrt{x_2(x_2 - 2)} \geq 0$ (always non-negative as a square root) For their sum to equal zero, both terms must individually equal zero: $$(x_1 - 1)^2 = 0 \quad \text{and} \quad \sqrt{x_2(x_2 - 2)} = 0$$ Step 4: Solve for $x_1$ and $x_2$. From $(x_1 - 1)^2 = 0$: $$x_1 = 1$$ From $\sqrt{x_2(x_2 - 2)} = 0$: $$x_2(x_2 - 2) = 0$$ $$x_2 = 0 \quad \text{or} \quad x_2 = 2$$ Since $x_2$ must be a positive real solution, we have: $$x_2 = 2$$ Step 5: Apply Vieta's formulas to find $b$ and $c$. For the quadratic $x^2 - bx + c = 0$ with roots $x_1$ and $x_2$, Vieta's formulas give: $$b = x_1 + x_2 = 1 + 2 = 3$$ $$c = x_1 \cdot x_2 = 1 \cdot 2 = 2$$ Step 6: Calculate the minimum value of $(b + c)$. $$b + c = 3 + 2 = 5$$ Since the constraint equation uniquely determines $x_1 = 1$ and $x_2 = 2$, there is only one possible value for $(b + c)$. **Final Answer:** The minimum value of $(b + c)$ is $\boxed{5}$, which corresponds to **Option 4**.