Multiple choice

If $4-5i$ is a root of the quadratic equation $x^2+ax+b=0$, then $(a, b)$ is equal to:

  1. $(8, 41)$
  2. $(-8, 41)$
  3. $(41, 8)$
  4. $(-41, 8)$
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B Correct answer
Explanation

For a quadratic equation with real coefficients, complex roots occur in conjugate pairs. If 4-5i is a root, 4+5i is also a root. Sum of roots = -a = (4-5i) + (4+5i) = 8, so a = -8. Product of roots = b = (4-5i)(4+5i) = 16 + 25 = 41.

AI explanation

Because a quadratic equation with real coefficients has complex conjugate roots, if 4-5i is a root, then 4+5i must also be a root. The sum of the roots is (4-5i) + (4+5i) = 8, which means a = -8. The product of the roots is (4-5i)(4+5i) = 16 + 25 = 41, meaning b = 41. Therefore, the pair (a, b) is (-8, 41).

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