Multiple choice

If $\left(3+i\right)$ is a root of the equation $x^{2}+ax+b=0$ then $a$ is

  1. $3$
  2. $-3$
  3. $6$
  4. $-6$
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D Correct answer
Explanation

For a quadratic equation with real coefficients, complex roots occur in conjugate pairs. If 3+i is a root, 3-i must also be a root. The sum of roots is (3+i) + (3-i) = 6. By Vieta's formulas, the sum of roots is -a. Thus, -a = 6, so a = -6.

AI explanation

For a quadratic equation with real coefficients, complex roots must occur in conjugate pairs, so the other root of x^2 + ax + b = 0 must be (3 - i). Using the sum of roots formula, the sum of the roots is (3 + i) + (3 - i) = 6. Since the sum of the roots equals -a for the standard quadratic equation, we set -a = 6. Solving for a gives a = -6. The result is -6.

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