1963 AMC 12 第 14 题

Question

给定方程

x^2+kx+6=0

和

x^2-kx+6=0

。如果适当排列两方程的根后，第二个方程的每个根都比第一个方程的对应根大

5

，那么

k

等于：

Given the equations

x^2+kx+6=0

and

x^2-kx+6=0.

If, when the roots of the equations are suitably listed, each root of the second equation is

5

more than the corresponding root of the first equation, then

k

equals:

5

-5

7

-7

以上都不是

none of these

比较两组根的和

Compare the sums of the two pairs of roots

每个根都增加

5

，会使根的和增加

10

Adding

5

to each root increases their sum by

10

第一个方程的两根之和为

-k

，第二个方程的两根之和为

k

。因为两个根都增加了

5

，所以

k=-k+10

，从而

k=5

。

所以正确答案是 A。

The first pair of roots has sum

-k,

while the second has sum

k.

Since both roots increase by

5,

k=-k+10,

so

k=5.

Therefore, the correct answer is A.

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