Question

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 $\begin{aligned} &(x-a)(x-b) \\ &\quad +(x-b)(x-c)=0? \end{aligned}$

$15$

$15.5$

$16$

$16.5$

$17$

Answer: D

Concepts:quadratic factoring optimization

Difficulty rating: 1280

Solution:

Factoring the left-hand side gives $(x-b)(2x-a-c)=0.$ Thus the roots are $b$ and $\frac{a+c}{2},$ whose sum is $b+\frac{a+c}{2}.$

The coefficient of $b$ in this sum is twice the coefficient of either $a$ or $c,$ so assign the largest digit to $b.$ The next two largest distinct digits should be $a$ and $c.$

Taking $b=9$ and $\{a,c\}=\{7,8\}$ gives $9+\frac{7+8}{2}=16.5.$

Thus, the correct answer is D .

2015 AMC 10B Problem 14

Answer: D

Difficulty rating: 1280

Solution:

Problem 14 in Other Years