2020 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 5 Easy

What is the sum of all real numbers x for which \left|x^{2}-12 x+34\right|=2?

  • A.

    12

  • B.

    15

  • C.

    18

  • D.

    21

  • E.

    25

Answer:C

Solution 1 ：

Split the equation into two cases, where the value inside the absolute value is positive and nonpositive. Case 1: The equation yields x^{2}-12 x+34=2, which is equal to (x-4)(x-8)=0. Therefore, the two values for the positive case is 4 and 8 . Case 2: Similarly, taking the nonpositive case for the value inside the absolute value notation yields -x^{2}+12 x-34=2. Factoring and simplifying gives (x-6)^{2}=0, so the only value for this case is 6 . Summing all the values results in 4+8+6=(\mathbf{C}) 18

Solution 2:

We have the equations x^{2}-12 x+32=0 and x^{2}-12 x+36=0. Notice that the second is a perfect square with a double root at x=6, and the first has real roots. By Vieta's, the sum of the roots of the first equation is 12.12+6= (C) 18

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