Multiple choice

If $a\pm ib$ are roots of the equation $x^2+ax+b$, then (where $i=\sqrt{-1}$)

  1. $a > 0$
  2. $a < 0$
  3. $a = 0$

  4. none of these
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C Correct answer
Explanation

If a + ib and a - ib are roots of x^2 + ax + b = 0, the sum of roots is (a + ib) + (a - ib) = 2a. From the equation, the sum of roots is -a. Thus, 2a = -a, which implies 3a = 0, so a = 0.

AI explanation

For the complex conjugate roots a+ib and a-ib, the sum of the roots is (a+ib)+(a-ib)=2a. The standard form x^2+ax+b=0 has a coefficient of 1 for x^2, so the sum of the roots also equals -a. Equating the two sums gives 2a=-a, which means 3a=0 and thus a=0. Therefore, a=0.

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