AMC10 2020 A

AMC10 2020 A · Q5

AMC10 2020 A · Q5. It mainly tests Absolute value, Quadratic equations.

What is the sum of all real numbers $x$ for which $|x^2 - 12x + 34| = 2$?

所有满足$|x^2 - 12x + 34| = 2$的实数$x$之和是多少？

(A) 12 12

(B) 15 15

(D) 21 21

(E) 25 25

Answer

Correct choice: (C)

正确答案：（C）

Solution

The given equation is equivalent to $x^2 - 12x + 34 = \pm 2$, which splits into two cases, $x^2 - 12x + 32 = 0$ and $x^2 - 12x + 36 = 0$. Because $x^2 - 12x + 32 = (x-4)(x-8)$, values of $x$ that satisfy the first equation are 4 and 8; and because $x^2 - 12x + 36 = (x-6)^2$, the only solution of the second equation is 6. The sum of these three values is $4 + 8 + 6 = 18$.

原方程等价于$x^2 - 12x + 34 = \pm 2$，分为两情况：$x^2 - 12x + 32 = 0$和$x^2 - 12x + 36 = 0$。因为$x^2 - 12x + 32 = (x-4)(x-8)$，第一方程解为4和8；因为$x^2 - 12x + 36 = (x-6)^2$，第二方程唯一解为6。这三个值的和是$4 + 8 + 6 = 18$。

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