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(D) Let the roots of $x^2 - bx + c = 0$ be $\alpha$ and $\beta$, and the roots of $x^2 - cx + b = 0$ be $\alpha'$ and $\beta'$.Given that the differen

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$-4$

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(D) Let the roots of $x^2 - bx + c = 0$ be $\alpha$ and $\beta$, and the roots of $x^2 - cx + b = 0$ be $\alpha'$ and $\beta'$.Given that the difference between the roots is the same, we have $|\alpha - \beta| = |\alpha' - \beta'|$.For the first equation, the difference of roots is $\alpha - \beta = \sqrt{(\alpha + \beta)^2 - 4\alpha\beta} = \sqrt{b^2 - 4c}$.For the second equation, the difference of roots is $\alpha' - \beta' = \sqrt{(\alpha' + \beta')^2 - 4\alpha'\beta'} = \sqrt{c^2 - 4b}$.Equating the squares of the differences: $b^2 - 4c = c^2 - 4b$.Rearranging the terms: $b^2 - c^2 = 4c - 4b$.Factoring the left side: $(b + c)(b - c) = -4(b - c)$.Assuming $b 
eq c$ (if $b = c$, the equations are identical), we can divide by $(b - c)$ to get $b + c = -4$.

If the roots of the equations $x^2 - bx + c = 0$ and $x^2 - cx + b = 0$ differ by the same quantity, then $b + c$ is equal to

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