AMC10 2015 B

AMC10 2015 B · Q14

AMC10 2015 B · Q14. It mainly tests Quadratic equations, Manipulating equations.

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$(x-a)(x-b)+(x-b)(x-c)=0$的根之和的最大值是多少？

(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 $9 + \frac{8+7}{2} = 16.5$.

若$(x-a)(x-b)+(x-b)(x-c) = 0$，则$(x-b)(2x-(a+c)) = 0$，因此两个根是$b$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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