How many real roots does the equation $x^2+3|x|+2=0$ have ?
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zero
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one
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Two
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Four
Reveal answer
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A
Correct answer
Explanation
For x^2 + 3|x| + 2 = 0, note that x^2 = |x|^2. The equation becomes |x|^2 + 3|x| + 2 = 0. Let u = |x|, then u^2 + 3u + 2 = 0, which factors to (u+1)(u+2) = 0. Since |x| must be non-negative, there are no real solutions for u = -1 or u = -2.
AI explanation
For any real value of x, the absolute value of x is non-negative, making both 3 times the absolute value of x and 2 strictly positive. When you add x squared, which is also non-negative, to these positive values, the sum x squared plus 3 times the absolute value of x plus 2 is always greater than zero. Since the expression can never equal zero, the equation has no real solutions. Therefore, the number of real roots is zero.