Key Concepts and Formulas

• Quadratic Formula: For a quadratic equation $ax^2 + bx + c = 0$, the roots are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• Complex Number Operations: Recall that $i = \sqrt{-1}$, so $i^2 = -1$. Complex numbers are added/subtracted by combining real and imaginary parts separately, and multiplied using the distributive property.
• Square Root of a Complex Number: If $\sqrt{a + bi} = p + qi$, then squaring both sides allows you to solve for $p$ and $q$ by equating real and imaginary parts.

Step-by-Step Solution

Step 1: Identify Coefficients

We are given the quadratic equation $x^2 - (3 - 2i)x - (2i - 2) = 0$. We identify the coefficients $a$, $b$, and $c$ by comparing to the standard form $ax^2 + bx + c = 0$: $a = 1$ $b = -(3 - 2i)$ $c = -(2i - 2)$ This step is crucial to correctly apply the quadratic formula in the following steps.

Step 2: Apply the Quadratic Formula

We substitute the coefficients into the quadratic formula: $x = \frac{-(-(3 - 2i)) \pm \sqrt{(-(3 - 2i))^2 - 4(1)(-(2i - 2))}}{2(1)}$ $x = \frac{(3 - 2i) \pm \sqrt{(3 - 2i)^2 + 4(2i - 2)}}{2}$ This step sets up the calculation of the roots using the known coefficients.

Step 3: Simplify the Discriminant

We simplify the expression under the square root, which is the discriminant, $\Delta$: $\Delta = (3 - 2i)^2 + 4(2i - 2)$ Expand the square term: $(3 - 2i)^2 = 3^2 - 2(3)(2i) + (2i)^2 = 9 - 12i + 4i^2 = 9 - 12i - 4 = 5 - 12i$. Substitute this back into the discriminant expression: $\Delta = (5 - 12i) + (8i - 8)$ $\Delta = (5 - 8) + (-12i + 8i)$ $\Delta = -3 - 4i$ So, the quadratic formula becomes: $x = \frac{3 - 2i \pm \sqrt{-3 - 4i}}{2}$ Simplifying the discriminant is necessary to find the roots.

Step 4: Find the Square Root of the Discriminant

We need to find $\sqrt{-3 - 4i}$. Let $\sqrt{-3 - 4i} = p + iq$, where $p$ and $q$ are real numbers. Squaring both sides: $(p + iq)^2 = -3 - 4i$ $p^2 + 2piq + (iq)^2 = -3 - 4i$ $p^2 - q^2 + 2piq = -3 - 4i$ Equating the real and imaginary parts:

1. $p^2 - q^2 = -3$
2. $2pq = -4 \Rightarrow pq = -2$

We also use the fact that $|p+iq|^2 = |-3-4i|$ which implies $p^2 + q^2 = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9+16} = \sqrt{25} = 5$. 3. $p^2 + q^2 = 5$

Now we have a system of two equations for $p^2$ and $q^2$: $p^2 - q^2 = -3$ $p^2 + q^2 = 5$

Adding these two equations: $(p^2 - q^2) + (p^2 + q^2) = -3 + 5$ $2p^2 = 2 \Rightarrow p^2 = 1 \Rightarrow p = \pm 1$

From equation (2), $pq = -2$. If $p = 1$, then $1 \cdot q = -2 \Rightarrow q = -2$. So, $1 - 2i$ is one square root. If $p = -1$, then $(-1) \cdot q = -2 \Rightarrow q = 2$. So, $-1 + 2i$ is the other square root. Thus, $\sqrt{-3 - 4i} = \pm (1 - 2i)$. Finding the square root allows us to finalize the application of the quadratic formula.

Step 5: Calculate the Roots

Now we substitute $\sqrt{-3 - 4i} = \pm (1 - 2i)$ back into the quadratic formula: $x = \frac{3 - 2i \pm (1 - 2i)}{2}$ This gives us two roots: $x_1 = \frac{(3 - 2i) + (1 - 2i)}{2} = \frac{4 - 4i}{2} = 2 - 2i$ $x_2 = \frac{(3 - 2i) - (1 - 2i)}{2} = \frac{3 - 2i - 1 + 2i}{2} = \frac{2}{2} = 1$ We have now calculated the two roots of the quadratic equation.

Step 6: Identify $\alpha, \beta, \gamma, \delta$

The problem states that the roots are $\alpha + i\beta$ and $\gamma + i\delta$. We can assign the calculated roots to these forms. The final expression $\alpha\gamma + \beta\delta$ is symmetric with respect to the roots, so the assignment order doesn't matter.

Let $z_1 = \alpha + i\beta = 2 - 2i$. Then $\alpha = 2$ and $\beta = -2$.

Let $z_2 = \gamma + i\delta = 1 = 1 + 0i$. Then $\gamma = 1$ and $\delta = 0$. Identifying the real and imaginary components is essential for the final calculation.

Step 7: Calculate $\alpha\gamma + \beta\delta$

Now, substitute the values of $\alpha, \beta, \gamma, \delta$ into the expression we need to find: $\alpha\gamma + \beta\delta = (2)(1) + (-2)(0)$ $\alpha\gamma + \beta\delta = 2 + 0$ $\alpha\gamma + \beta\delta = 2$ We have now calculated the required expression.

Common Mistakes & Tips

• Sign Errors: Be extremely careful with signs, especially when expanding squares and distributing negatives. A single sign error can lead to completely wrong roots.
• Complex Number Arithmetic: Remember that $i^2 = -1$. Make sure to correctly handle complex number arithmetic throughout the solution.
• Alternative Approach (Vieta's): While not strictly necessary here, remember Vieta's formulas. If you need to relate roots without explicitly solving, they can be faster. Also, use Vieta's to check the correctness of your roots.

Summary

This problem required us to solve a quadratic equation with complex coefficients, involving the use of the quadratic formula and the process of finding the square root of a complex number. By carefully applying these techniques, we found the roots and then calculated the value of the expression $\alpha\gamma + \beta\delta$, which is equal to 2.

Final Answer

The final answer is $\boxed{2}$, which corresponds to option (A).