1967 AMC 12 第 35 题

Question

方程

64x^3-144x^2+92x-15=0

的各根成等差数列。最大根与最小根之差为：

The roots of

64x^3-144x^2+92x-15=0

are in arithmetic progression. The difference between the largest and smallest roots is:

2

1

\dfrac12

\dfrac38

\dfrac14

将三个根写成

t-d

、

t

、

t+d

Write the roots as

t-d,

t,

and

t+d

先利用根的和求出

t

，再利用根的积求出

d^2

Use their sum to find

t,

then use their product to find

d^2

设三个根为

t-d

、

t

、

t+d

。它们的和为

3t=\frac{144}{64}=\frac{9}{4}

，所以

t=\frac{3}{4}

。它们的积为

t(t^2-d^2)=\frac{15}{64}\text{。}

代入

t=\frac{3}{4}

得

d^2=\frac{1}{4}

，所以最大根与最小根之差为

2\lvert d\rvert=1

。

因此，正确答案是 B。

Let the roots be

t-d,

t,

and

t+d.

Their sum is

3t=\frac{144}{64}=\frac{9}{4},

so

t=\frac{3}{4}.

Their product is

t(t^2-d^2)=\frac{15}{64}.

Substituting

t=\frac{3}{4}

gives

d^2=\frac{1}{4},

so the difference between the extreme roots is

2\lvert d\rvert=1.

Therefore, the correct answer is B.

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