Key Concepts and Formulas

• Roots of a Quadratic Equation: If $\alpha$ is a root of the equation $ax^2 + bx + c = 0$, then $a\alpha^2 + b\alpha + c = 0$.
• Modulus of a Complex Number: For a complex number $z = a + bi$, its modulus is given by $|z| = \sqrt{a^2 + b^2}$.
• Properties of Modulus: $|z^n| = |z|^n$ and $|kz| = |k||z|$ for any integer $n$ and real number $k$.

Step-by-Step Solution

Step 1: Analyze the Given Equation and its Roots

We are given the quadratic equation: $x^2 + (2i - 1) = 0$ We can rewrite this as $x^2 = 1 - 2i$ Since $\alpha$ and $\beta$ are the roots of the equation, they must satisfy the equation. Therefore, $\alpha^2 = 1 - 2i$ and $\beta^2 = 1 - 2i$ This tells us that $\alpha^2 = \beta^2$. This is a crucial relationship that will simplify our calculations.

Step 2: Calculate the Modulus of $\alpha^2$ (or $\beta^2$)

Since $\alpha^2 = 1 - 2i$, we can find its modulus. Let $z = 1 - 2i$. Then the real part is $a = 1$ and the imaginary part is $b = -2$. Using the formula for the modulus of a complex number, $|z| = \sqrt{a^2 + b^2}$: $|\alpha^2| = |1 - 2i| = \sqrt{(1)^2 + (-2)^2}$ $|\alpha^2| = \sqrt{1 + 4}$ $|\alpha^2| = \sqrt{5}$ Since $\alpha^2 = \beta^2$, we also have $|\beta^2| = \sqrt{5}$.

Step 3: Simplify the Expression $|\alpha^8 + \beta^8|$

We want to find the value of $|\alpha^8 + \beta^8|$. Since $\alpha^2 = \beta^2$, we have $(\alpha^2)^4 = (\beta^2)^4$, which means $\alpha^8 = \beta^8$ Now, substitute $\beta^8$ with $\alpha^8$ in the expression: $|\alpha^8 + \beta^8| = |\alpha^8 + \alpha^8|$ $|\alpha^8 + \beta^8| = |2\alpha^8|$ Using the modulus property $|kz| = |k||z|$, where $k=2$: $|2\alpha^8| = 2|\alpha^8|$ Using the modulus property $|z^n| = |z|^n$, we can write $|\alpha^8|$ as $|(\alpha^2)^4|$ and then as $|\alpha^2|^4$: $2|\alpha^8| = 2|(\alpha^2)^4|$ $2|(\alpha^2)^4| = 2|\alpha^2|^4$

Step 4: Substitute and Compute the Final Result

From Step 2, we found that $|\alpha^2| = \sqrt{5}$. Substituting this value into our simplified expression: $2|\alpha^2|^4 = 2(\sqrt{5})^4$ To compute $(\sqrt{5})^4$: $(\sqrt{5})^4 = (5^{1/2})^4$ $= 5^{(1/2) \times 4}$ $= 5^2$ $= 25$ Finally, multiply by 2: $2 \times 25 = 50$ Thus, the value of $|\alpha^8 + \beta^8|$ is $50$.

Common Mistakes & Tips

• Avoid Explicit Root Calculation: Avoid the unnecessary step of finding the exact values of $\alpha$ and $\beta$. Instead, utilize the properties of roots and moduli.
• Correct Modulus Calculation: Ensure correct calculation of the modulus. $|a+bi| = \sqrt{a^2+b^2}$ is the correct formula.
• Use Modulus Properties: Remember and apply the properties of modulus, such as $|z^n| = |z|^n$ and $|kz| = |k||z|$.

Summary

This problem demonstrates the power of using properties of complex numbers and the definition of roots to simplify calculations. The key insight was recognizing that $\alpha^2 = \beta^2$, which significantly simplified the expression $|\alpha^8 + \beta^8|$ and eliminated the need to find the actual complex roots. The final answer is 50.

Final Answer The final answer is $\boxed{50}$, which corresponds to option (A).