AMC12 2000 A

AMC12 2000 A · Q15

AMC12 2000 A · Q15. It mainly tests Quadratic equations, Functions basics.

Let $f$ be a function for which $f\left(\dfrac{x}{3}\right) = x^2 + x + 1$. Find the sum of all values of $z$ for which $f(3z) = 7$.

设函数 $f$ 满足 $f\left(\dfrac{x}{3}\right) = x^2 + x + 1$。求所有满足 $f(3z) = 7$ 的 $z$ 的和。

(A) $-1/3$ $-1/3$

(B) $-1/9$ $-1/9$

(D) $5/9$ $5/9$

(E) $5/3$ $5/3$

Answer

Correct choice: (B)

正确答案：（B）

Solution

Let $y = \frac{x}{3}$; then $f(y) = (3y)^2 + 3y + 1 = 9y^2 + 3y+1$. Thus $f(3z)-7=81z^2+9z-6=3(9z-2)(3z+1)=0$, and $z = -\frac{1}{3}, \frac{2}{9}$. These sum up to $\boxed{\textbf{(B) }-\frac19}$.

令 $y = \frac{x}{3}$；则 $f(y) = (3y)^2 + 3y + 1 = 9y^2 + 3y+1$。因此 $f(3z)-7=81z^2+9z-6=3(9z-2)(3z+1)=0$，从而 $z = -\frac{1}{3}, \frac{2}{9}$。它们的和为 $\boxed{\textbf{(B) }-\frac19}$。

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