2005 AMC 10A 第 10 题

Question

有两个

a

的值使方程

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

只有一个

x

的解。这两个

a

的值之和是多少？

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

二次方程恰好有一个解时，判别式等于

0

。

A quadratic has exactly one solution when its discriminant equals

0

合并一次项为

(a + 8)x

，再令

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

。

Combine the linear terms into

(a + 8)x,

then set

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

方程可写为

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

，它只有一个解当且仅当判别式

(a+8)^2 - 144 = 0

。所以

a + 8 = \pm 12

，即

a = 4

或

a = -20

，二者之和为

-16

。

所以正确答案是 A。

Writing the equation as

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

there is one solution exactly when the discriminant

(a+8)^2 - 144 = 0.

Then

a + 8 = \pm 12,

so

a = 4

or

a = -20,

and their sum is

-16.

Thus, the correct answer is A.

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