Multiple choice

How many real roots does the quadratic equation $f\left( x \right) ={ x }^{ 2 }+3\left| x \right| +2=0$ have?

  1. One

  2. Two

  3. Four

  4. No real root
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Since x^2 = |x|^2, the equation is |x|^2 + 3|x| + 2 = 0. Let y = |x|, then y^2 + 3y + 2 = 0, which factors to (y+1)(y+2) = 0. This gives y = -1 or y = -2. Since y = |x| must be non-negative, there are no real solutions for x.

AI explanation

The equation f(x) = x^2 + 3|x| + 2 = 0 has no real roots because the sum of squared terms and absolute values cannot equal zero for real numbers. Specifically, since x^2 is always non-negative and |x| is always non-negative, the minimum value of the expression occurs at x = 0, which equals 2, not 0. Therefore, there are no real roots.

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