If + i and + i are the roots of the equation x^2 - (3-2i)x - (2i-2) = 0, i = -1, then + is equal to:

If + i and + i are the roots of the equation x^2 - (3-2i)x - (2i-… — Solution | JEE Mathematics

MCQMediumJEE 2025Quadratic Equations in Complex Numbers

If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of the equation $x^2 - (3-2i)x - (2i-2) = 0$ , $i = \sqrt{-1}$ , then $\alpha\gamma + \beta\delta$ is equal to:

A
$6$
B
$2$
C
$-2$
D
$-6$

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given: The roots are $\alpha + i\beta$ and $\gamma + i\delta$ for the equation

x^2 - (3 - 2i)x - (2i - 2) = 0

Find: $\alpha\gamma + \beta\delta$

Using Vieta's relations for the quadratic $x^2 - (3 - 2i)x - (2i - 2) = 0$ , the product of the roots is

(\alpha + i\beta)(\gamma + i\delta) = -(2i - 2) = 2 - 2i

Now expand the product:

(\alpha + i\beta)(\gamma + i\delta) = (\alpha\gamma - \beta\delta) + i(\alpha\delta + \beta\gamma)

Comparing with $2 - 2i$ gives

\alpha\gamma - \beta\delta = 2

The provided the solution concludes with Option B. It also lists roots $2 - 2i$ and $1 + 0i$ , for which

\alpha = 2,\; \beta = -2,\; \gamma = 1,\; \delta = 0

Hence

\alpha\gamma + \beta\delta = 2 \cdot 1 + (-2) \cdot 0 = 2

Therefore, the correct option is B.

Using the roots stated in the solution

Given: the solution states the roots are $2 - 2i$ and $1 + 0i$ . Find: $\alpha\gamma + \beta\delta$

Match the roots with the form $\alpha + i\beta$ and $\gamma + i\delta$ :

\alpha + i\beta = 2 - 2i \Rightarrow \alpha = 2,\; \beta = -2

\gamma + i\delta = 1 + 0i \Rightarrow \gamma = 1,\; \delta = 0

Now compute:

\alpha\gamma + \beta\delta = 2 \cdot 1 + (-2) \cdot 0 = 2

Therefore, $\alpha\gamma + \beta\delta = 2$ , so the correct option is B.

Common mistakes

Using the product of roots incorrectly. For $x^2 + bx + c = 0$ , the product is $c$ , but here the constant term is $-(2i-2) = 2-2i$ . Always rewrite the quadratic carefully before applying Vieta's relations.
Confusing $\alpha\gamma + \beta\delta$ with the real part of $(\alpha + i\beta)(\gamma + i\delta)$ . The real part is $\alpha\gamma - \beta\delta$ , not the required expression. Read the target expression separately before comparing parts.
Assigning the roots to $\alpha, \beta, \gamma, \delta$ incorrectly. If a root is $1 + 0i$ , then $\delta = 0$ must be used explicitly. Do not ignore the imaginary part when extracting coefficients.

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Answer

Step-by-step solution

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