Multiple choice

If one root of the equation $\displaystyle { x }^{ 2 }+\left( 1-3i \right) x-2\left( 1+i \right) =0$ is $\displaystyle -1+i$, then the other root is

  1. $\displaystyle -1-i$
  2. $\displaystyle \frac { \left( -1-i \right) }{ 2 } $
  3. $\displaystyle i$
  4. $\displaystyle 2i$
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D Correct answer
Explanation

For a quadratic equation with complex coefficients, the roots do not necessarily come in conjugate pairs. Using the sum of roots: (-1+i) + r2 = -(1-3i) = -1+3i. r2 = -1+3i - (-1+i) = 2i.

AI explanation

For a quadratic equation with real coefficients, complex roots must appear as conjugate pairs, but since the coefficient (1 minus 3i) is complex, the conjugate root theorem does not apply. We use Vieta's formulas, which state that the sum of the roots equals negative the coefficient of x. The sum of the two roots is negative the quantity (1 minus 3i). Since one root is given as negative 1 plus i, we subtract it from the sum to find the other root. Therefore, the second root equals negative 1 plus 3i minus negative 1 plus i, which simplifies to 2i.

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