2005 AMC 12A 第 9 题

Question

有两个

a

的值使方程

4x^2 + ax + 8x + 9 = 0

对

x

只有一个解。这两个

a

的和是多少？

There are two values of

a

for which the equation

4x^2 + ax + 8x + 9 = 0

has only one solution for

x.

What is the sum of those values of

a?

-16

-8

0

8

20

二次方程只有一个解，当且仅当判别式为

0

。

A quadratic has one solution exactly when its discriminant is

0

把一次项系数写成

a + 8

，令

(a+8)^2 - 4 \cdot 4 \cdot 9 = 0

。

Write the linear coefficient as

a + 8

and set

(a+8)^2 - 4 \cdot 4 \cdot 9 = 0

方程为

4x^2 + (a+8)x + 9 = 0

。它只有一个解时判别式为零：

(a+8)^2 - 4 \cdot 4 \cdot 9 = 0\text{，}

所以

(a+8)^2 = 144

，

a + 8 = \pm 12

。

因此

a = 4

或

a = -20

，和为

-16

。

所以正确答案是 A。

The equation is

4x^2 + (a+8)x + 9 = 0.

It has one solution when the discriminant vanishes:

(a+8)^2 - 4 \cdot 4 \cdot 9 = 0,

so

(a+8)^2 = 144

and

a + 8 = \pm 12.

Thus

a = 4

or

a = -20,

and their sum is

-16.

Thus, the correct answer is A.

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