2020 AMC 10A Problem 5

Question

What is the sum of all real numbers

x

for which

|x^2-12x+34|=2?

12

15

18

21

25

Split the absolute-value equation into two quadratic equations

Remember to count the double root only once in the sum of real numbers

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The equation means

x^2-12x+34=2

or

x^2-12x+34=-2

. The first gives

x^2-12x+32=0

, with roots

4

and

8

. The second gives

(x-6)^2=0

, with root

6

. The sum of all real solutions is

4+8+6=18

. Thus, C is the correct answer.

Problem 5 in other years