2015 AMC 12B Problem 12

Below is the professionally curated solution for Problem 12 of the 2015 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2015 AMC 12B solutions, or check the answer key.

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Concepts:factoring optimization

Difficulty rating: 1540

12.

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 $(x - a)(x - b) + (x - b)(x - c) = 0?$

$15$

$15.5$

$16$

$16.5$

$17$

Solution:

Factoring gives $(x - b)\bigl(2x - (a + c)\bigr) = 0,$ so the roots are $b$ and $\dfrac{a + c}{2}.$ Their sum is $b + \dfrac{a + c}{2}.$

Using distinct digits, take $b = 9$ and $a + c = 8 + 7 = 15,$ giving $9 + 7.5 = 16.5.$

Thus, the correct answer is D.

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