Key Concepts and Formulas

• Conjugate Root Theorem: If a polynomial equation with real coefficients has a complex root $a + bi$, then its conjugate $a - bi$ is also a root.
• Vieta's Formulas: For a quadratic equation $ax^2 + bx + c = 0$, the sum of the roots is $-\frac{b}{a}$ and the product of the roots is $\frac{c}{a}$.
• Complex Number Arithmetic: Remember that $i^2 = -1$ when performing operations with complex numbers.

Step-by-Step Solution

1. Identify the Roots of the Quadratic Equation

• What and Why: We are given one root of the quadratic equation and told that the coefficients are real. We use the Conjugate Root Theorem to find the other root.
• Calculation: Since $1 - 2i$ is a root and the coefficients $\alpha$ and $\beta$ are real, the conjugate $1 + 2i$ is also a root. Therefore, the roots are $z_1 = 1 - 2i$ and $z_2 = 1 + 2i$.
• Reasoning: The Conjugate Root Theorem applies because the problem states $\alpha, \beta \in \mathbb{R}$.

2. Apply Vieta's Formulas to Find $\alpha$ and $\beta$

• What and Why: We use Vieta's formulas to relate the roots of the quadratic equation to its coefficients. This allows us to find the values of $\alpha$ and $\beta$.
• 2a. Calculate the Sum of Roots to find $\alpha$
  • Calculation: $z_1 + z_2 = (1 - 2i) + (1 + 2i) = 2$.
  • Reasoning: According to Vieta's formulas, the sum of the roots is equal to $-\alpha$.
  • Calculation: Therefore, $-\alpha = 2$, which implies $\alpha = -2$.
• 2b. Calculate the Product of Roots to find $\beta$
  • Calculation: $z_1 \cdot z_2 = (1 - 2i)(1 + 2i) = 1^2 - (2i)^2 = 1 - 4i^2 = 1 - 4(-1) = 1 + 4 = 5$.
  • Reasoning: According to Vieta's formulas, the product of the roots is equal to $\beta$.
  • Calculation: Therefore, $\beta = 5$.

3. Calculate the Value of $\alpha - \beta$

• What and Why: Now that we have found $\alpha$ and $\beta$, we can calculate the value of the expression $\alpha - \beta$.
• Calculation: $\alpha - \beta = -2 - 5 = -7$.
• Reasoning: Simple subtraction using the values we found for $\alpha$ and $\beta$.

Common Mistakes & Tips

• Forgetting the Conjugate: Always remember to take the conjugate when using the Conjugate Root Theorem. A common mistake is to forget to change the sign of the imaginary part.
• Sign Errors with Vieta's: Be careful with the signs in Vieta's formulas. The sum of the roots is equal to negative $b/a$.
• Incorrect $i^2$ Value: Remember that $i^2 = -1$. This is a fundamental property of complex numbers.

Summary

We used the Conjugate Root Theorem to find both roots of the quadratic equation. Then, we applied Vieta's formulas to relate the roots to the coefficients $\alpha$ and $\beta$. Finally, we calculated $\alpha - \beta$ to find the answer.

The final answer is \boxed{-7}, which corresponds to option (A).