If α + iβ and γ + iδ are the roots of x² - (3 - 2i)x - (2i - 2) = 0, i = √-1, then α γ + β δ is equal to:

Solution

Maths sir · 4 steps

The coefficients are complex, so the roots need not be conjugates. Find both roots outright.

1. Discriminant. Here b = -(3-2i) and c = 2-2i. So D = (3-2i)² - 4(2-2i) = (5-12i) - (8-8i) = -3-4i.
2. Square root. Try (a+bi)² = -3-4i, which needs a² - b² = -3 and 2ab = -4. The pair a = 1, b = -2 fits, so √D = 1-2i.

   x = ((3-2i) ± (1-2i))/2 ⇒ x = 2-2i or x = 1

3. Check. The sum is 3-2i and the product is (2-2i)(1) = 2-2i, which equals -(2i-2). Both match the equation.
4. So. Take α = 2, β = -2, γ = 1, δ = 0. Then αγ + βδ = 2 + 0 = 2. Swapping the two roots gives the same value, so the answer is (a).

The idea to keep. Roots come in conjugate pairs only when the coefficients are real. With complex coefficients, use the quadratic formula and find √D by solving (a+bi)² = D.

Answer from the NTA final answer key. Solution by PrepWiser.