AMC12 2015 B

AMC12 2015 B · Q12

AMC12 2015 B · Q12. It mainly tests Quadratic equations, Vieta / quadratic relationships (basic).

Let $a$, $b$, and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $(x - a)(x - b) + (x - b)(x - c) = 0$?

设 $a$、$b$ 和 $c$ 是三个不同的个位数。方程 $(x - a)(x - b) + (x - b)(x - c) = 0$ 的根的和的最大值是多少？

(A) 15 15

(B) 15.5 15.5

(D) 16.5 16.5

(E) 17 17

Answer

Correct choice: (D)

正确答案：（D）

Solution

If $(x-a)(x-b) + (x-b)(x-c) = 0$, then $(x-b)(2x - (a+c)) = 0$, so the two roots are $b$ and $\frac{a+c}{2}$. The maximum value of their sum is $9 + \frac{8+7}{2} = 16.5$.

如果 $(x-a)(x-b) + (x-b)(x-c) = 0$，则 $(x-b)(2x - (a+c)) = 0$，所以两个根是 $b$ 和 $\frac{a+c}{2}$。它们的和的最大值是 $9 + \frac{8+7}{2} = 16.5$。

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