AMC12 2018 A

AMC12 2018 A · Q5

AMC12 2018 A · Q5. It mainly tests Quadratic equations, Factoring.

What is the sum of all possible values of $k$ for which the polynomials $x^{2} -3x + 2$ and $x^{2} -5x + k$ have a root in common?

对于哪些$k$的值，多项式$x^{2} -3x + 2$和$x^{2} -5x + k$有公共根？所有可能$k$之和是多少？

(A) 3 3

(B) 4 4

(D) 6 6

(E) 10 10

Answer

Correct choice: (E)

正确答案：（E）

Solution

Answer (E): Factoring $x^2-3x+2$ as $(x-1)(x-2)$ shows that its roots are 1 and 2. If 1 is a root of $x^2-5x+k$, then $1^2-5\cdot1+k=0$ and $k=4$. If 2 is a root of $x^2-5x+k$, then $2^2-5\cdot2+k=0$ and $k=6$. The sum of all possible values of $k$ is $4+6=10$.

答案（E）：将 $x^2-3x+2$ 因式分解为 $(x-1)(x-2)$ 可知其根为 1 和 2。若 1 是 $x^2-5x+k$ 的根，则 $1^2-5\cdot1+k=0$，所以 $k=4$。若 2 是 $x^2-5x+k$ 的根，则 $2^2-5\cdot2+k=0$，所以 $k=6$。所有可能的 $k$ 值之和为 $4+6=10$。

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