1951 AMC 12 第 16 题

Question

对二次方程

f(x)=ax^2+bx+c=0

使用求根公式时，若恰有

c=\dfrac{b^2}{4a}

，则

y=f(x)

的图像一定：

If in applying the quadratic formula to a quadratic equation

f(x)=ax^2+bx+c=0,

it happens that

c=\dfrac{b^2}{4a},

then the graph of

y=f(x)

will certainly:

有最大值

Have a maximum

有最小值

Have a minimum

与

x

轴相切

Be tangent to the

x

-axis

与

y

轴相切

Be tangent to the

y

-axis

只位于一个象限内

Lie in one quadrant only

将所给条件代入判别式

b^2-4ac

Substitute the condition into the discriminant

b^2-4ac

有一个二重实根的二次函数图像只在一点接触横轴

A quadratic with one repeated real root touches the horizontal axis once

由所给条件可得

b^2-4ac=b^2-4a\left(\frac{b^2}{4a}\right)=0\text{。}

因此，该二次方程有一个二重实根，其抛物线与

x

轴相切。

因此，正确答案是 C。

The condition gives

b^2-4ac=b^2-4a\left(\frac{b^2}{4a}\right)=0.

Hence the quadratic has a repeated real root, so its parabola is tangent to the

x

-axis.

Thus, the correct answer is C.

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