Multiple choice

The number of roots of the equation $|x|^2-7|x|+12=0$ is?

  1. $1$
  2. $2$
  3. $3$
  4. $4$
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D Correct answer
Explanation

Let y = |x|. The equation is y^2 - 7y + 12 = 0, which factors to (y - 3)(y - 4) = 0. Thus |x| = 3 or |x| = 4. This gives x = 3, -3, 4, -4, resulting in 4 roots.

AI explanation

Treat the absolute value of x as a single variable, y, where y is positive. The equation becomes the quadratic y^2 - 7y + 12 = 0, which factors into (y - 3)(y - 4) = 0, giving roots y = 3 and y = 4. Since the absolute value of x is 3 and 4, x can be 3, -3, 4, or -4. This gives a total of four roots.

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