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Step-by-Step Solution
Step 1: Identify the Conjugate Root
Since the given quadratic equation has real coefficients (i.e., $ \alpha $ and $ \beta $ are real), the complex roots come in conjugate pairs. Hence, if $1 - 2i$ is a root, then its conjugate $1 + 2i$ is also a root.
Step 2: Use the Sum of Roots
For a quadratic equation $z^2 + \alpha z + \beta = 0$, if the roots are $z_1$ and $z_2$, then:
$$ z_1 + z_2 = -\alpha. $$
Here, the roots are $1 - 2i$ and $1 + 2i$, so their sum is:
$$ (1 - 2i) + (1 + 2i) = 2. $$
Therefore,
$$ 2 = -\alpha \quad \Longrightarrow \quad \alpha = -2. $$
Step 3: Use the Product of Roots
For the same quadratic equation, the product of the roots is:
$$ z_1 \cdot z_2 = \beta. $$
Hence,
$$ (1 - 2i)\,(1 + 2i) = 1 - (2i)^2 = 1 - 4(-1) = 1 + 4 = 5. $$
Thus,
$$ \beta = 5. $$
Step 4: Compute $ \alpha - \beta $
We have:
$$ \alpha = -2 \quad \text{and} \quad \beta = 5. $$
Therefore,
$$ \alpha - \beta = -2 - 5 = -7. $$
Final Answer
The value of $ \alpha - \beta $ is $ -7.$
