Multiple choice

The equation $\displaystyle { x }^{ 2 }-3\left| x \right| +2=0$ has

  1. No real roots

  2. One real root

  3. Two real roots

  4. Four real roots
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let |x| = y. The equation becomes y^2 - 3y + 2 = 0, which factors to (y-1)(y-2) = 0. So |x| = 1 or |x| = 2. This gives x = 1, -1, 2, -2, totaling four real roots.

AI explanation

Because x squared has the same value as the absolute value of x squared, we can rewrite the equation as the absolute value of x squared minus 3 times the absolute value of x plus 2 equals 0. Factoring this quadratic in terms of the absolute value of x gives (the absolute value of x minus 2) times (the absolute value of x minus 1) equals 0. This provides two positive solutions for the absolute value of x: 2 and 1. Since the absolute value of x can be either 2 or 1, the original variable x can take four possible real values: 2, minus 2, 1, and minus 1.

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