1971 AMC 12 第 20 题

Question

方程

x^2+2hx=3

的两根平方和为

10

。

h

的绝对值等于：

The sum of the squares of the roots of the equation

x^2+2hx=3

is

10.

The absolute value of

h

is equal to:

-1

\frac{1}{2}

\frac{2}{3}

2

以上都不是

None of these

利用两根的和与积

Use the sum and product of the two roots

若两根为

r,s

，则

r+s=-2h

且

rs=-3

If the roots are

r,s,

then

r+s=-2h

and

rs=-3

设两根为

r,s

，由韦达定理，

r+s=-2h

且

rs=-3

。于是

\begin{aligned} 4h^2=(r+s)^2 &=r^2+s^2+2rs\\ &=10-6=4\text{。} \end{aligned}

因此

|h|=1

。选项 A 为

-1

，所以选项 A 至 D 中均没有该值。

因此，正确答案为 E。

For roots

r,s,

Vieta’s formulas give

r+s=-2h

and

rs=-3.

Hence

\begin{aligned} 4h^2=(r+s)^2 &=r^2+s^2+2rs\\ &=10-6=4. \end{aligned}

Therefore

|h|=1,

which is not among choices A-D because choice A is

-1.

Therefore, the correct answer is E.

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