Question

We may say concerning the solution of $|x|^2+|x|-6=0$ that:

there is only one root

the sum of the roots is $1$

the sum of the roots is $0$

the product of the roots is $4$

the product of the roots is $-6$

Answer: C

Concepts:absolute value quadratic zero product property

Difficulty rating: 1280

Small Hint:

Set $u=|x|$ , so $u\ge0$

Big Hint:

Factor the resulting quadratic in $u$ , then translate the admissible value back to $x$

Solution:

Let $u=|x|\ge0.$ Then $\begin{aligned} u^2+u-6&=0,\\ (u+3)(u-2)&=0. \end{aligned}$ The nonnegative solution is $u=2,$ so $|x|=2$ and $x=\pm2.$ Their sum is $0.$

Therefore, the correct answer is C.

1958 AMC 12 Problem 39

Answer: C

Difficulty rating: 1280

Small Hint:

Big Hint:

Solution:

Problem 39 in Other Years