Multiple choice

Find the sum of the real roots of the equation $x^2+ 5|x|+ 6=0$

  1. $5$
  2. $10$
  3. $-5$

  4. None of these
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D Correct answer
Explanation

The equation x^2 + 5|x| + 6 = 0 can be written as |x|^2 + 5|x| + 6 = 0. Let u = |x|, then u^2 + 5u + 6 = 0, which factors to (u+2)(u+3) = 0. This gives |x| = -2 or |x| = -3. Since the absolute value cannot be negative, there are no real roots.

AI explanation

We analyze the absolute value equation x^2 + 5|x| + 6 = 0 by considering two cases. If x is positive or zero, the equation is x^2 + 5x + 6 = 0, which factors to (x+2)(x+3) = 0, giving roots x = -2 and x = -3, neither of which are positive. If x is negative, the equation is x^2 - 5x + 6 = 0, which factors to (x-2)(x-3) = 0, giving roots x = 2 and x = 3, neither of which are negative. Therefore, the equation has no real roots, making the sum of the real roots undefined.

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