1956 AMC 12 第 23 题

Question

方程

ax^2-2x\sqrt2+c=0

中，

a

和

c

为实常数，且已知判别式为零。它的根必定：

About the equation

ax^2-2x\sqrt2+c=0,

with

a

and

c

real constants, we are told that the discriminant is zero. The roots are necessarily:

相等且为整数

equal and integral

相等且为有理数

equal and rational

相等且为实数

equal and real

相等且为无理数

equal and irrational

相等且为虚数

equal and imaginary

判别式为零使求根公式所得的两个值重合

A zero discriminant makes the two quadratic-formula values coincide

重根为

-\frac{b}{2a}=\frac{\sqrt2}{a}

The repeated root is

-\frac{b}{2a}=\frac{\sqrt2}{a}

实系数且判别式为零的二次方程有重根

x=\frac{-(-2\sqrt2)}{2a}=\frac{\sqrt2}{a}\text{。}

由于

a

是非零实数，此根为实数。它不一定总是有理数，也不一定总是无理数。

因此，两根必定相等且为实数，正确答案是 C。

A quadratic with real coefficients and discriminant zero has the repeated root

x=\frac{-(-2\sqrt2)}{2a}=\frac{\sqrt2}{a}.

Since

a

is a nonzero real number, this root is real. It need not always be rational or always be irrational.

Thus, the roots are necessarily equal and real, so the correct answer is C.

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