2020 AMC 10A 第 5 题

Question

所有满足下式的实数

x

之和是多少？

|x^2-12x+34|=2\text{？}

What is the sum of all real numbers

x

for which

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

12

15

18

21

25

把绝对值方程分成两个二次方程。

Split the absolute-value equation into two quadratic equations

对所有实数解求和时，重根只计算一次。

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

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方程等价于

x^2-12x+34=2

或

x^2-12x+34=-2

。第一式化为

x^2-12x+32=0

，根为

4

、

8

；第二式化为

(x-6)^2=0

，根为

6

。所有实数解的和为

4+8+6=18

。正确答案是 C。

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.

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