Key Concepts and Formulas

• Complex Conjugate: For a complex number $z = x + iy$, where $x, y \in \mathbb{R}$, the complex conjugate is $\bar{z} = x - iy$.
• Modulus of a Complex Number: The modulus of $z = x + iy$ is $|z| = \sqrt{x^2 + y^2}$. Also, $|z|^2 = z\bar{z}$.
• Properties of Conjugates: $\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$, $\overline{z^n} = (\bar{z})^n$, and $\overline{|z|} = |z|$.

Step-by-Step Solution

Step 1: Taking the Conjugate of the Given Equation

We are given the equation $(\bar{z})^2 + |z| = 0$. To obtain another equation relating $z$ and $\bar{z}$, we take the conjugate of both sides. This is a useful strategy when dealing with equations involving both $z$ and $\bar{z}$. $\overline{(\bar{z})^2 + |z|} = \overline{0}$ Using the properties of conjugates: $\overline{(\bar{z})^2} + \overline{|z|} = 0$ $(\overline{\bar{z}})^2 + |z| = 0$ Since $\overline{\bar{z}} = z$, we have: $z^2 + |z| = 0 \quad \text{... (2)}$

Step 2: Equating the Squared Terms

We now have two equations:

1. $(\bar{z})^2 + |z| = 0$
2. $z^2 + |z| = 0$

From these equations, we can express the squared terms in terms of $|z|$: $(\bar{z})^2 = -|z|$ and $z^2 = -|z|$. Therefore, we can equate the squared terms: $(\bar{z})^2 = z^2$

Step 3: Factoring the Equation and Deriving Conditions

Rearrange the equation $(\bar{z})^2 = z^2$ to get: $z^2 - (\bar{z})^2 = 0$ Factor the difference of squares: $(z - \bar{z})(z + \bar{z}) = 0$ This implies either $z - \bar{z} = 0$ or $z + \bar{z} = 0$. We now consider these two cases separately.

Case A: $z - \bar{z} = 0 \implies z = \bar{z}$ Case B: $z + \bar{z} = 0 \implies z = -\bar{z}$

Step 4: Analyzing Case A: $z = \bar{z}$

Let $z = x + iy$, where $x, y \in \mathbb{R}$. If $z = \bar{z}$, then $x + iy = x - iy$, which implies $2iy = 0$, so $y = 0$. Thus, $z = x$, which means $z$ is a real number. Substitute $z = x$ into the original equation $(\bar{z})^2 + |z| = 0$: $(\bar{x})^2 + |x| = 0$ Since $x$ is real, $\bar{x} = x$, so $x^2 + |x| = 0$ Since $x^2 \ge 0$ and $|x| \ge 0$, the only solution is $x = 0$. Thus, $z = 0$. However, we are looking for non-zero solutions, so we discard this solution.

Step 5: Analyzing Case B: $z = -\bar{z}$

Let $z = x + iy$. If $z = -\bar{z}$, then $x + iy = -(x - iy)$, which implies $x + iy = -x + iy$, so $2x = 0$, and $x = 0$. Thus, $z = iy$, which means $z$ is a purely imaginary number. Substitute $z = iy$ into the original equation $(\bar{z})^2 + |z| = 0$: $(\overline{iy})^2 + |iy| = 0$ $(-iy)^2 + |iy| = 0$ $(-i)^2 y^2 + |y| = 0$ $-y^2 + |y| = 0$ $|y| = y^2$ Since we are looking for non-zero solutions, $y \neq 0$. If $y > 0$, then $|y| = y$, so $y = y^2$, which means $y^2 - y = 0$, so $y(y - 1) = 0$. Since $y \neq 0$, we have $y = 1$. Thus, $z = i$. If $y < 0$, then $|y| = -y$, so $-y = y^2$, which means $y^2 + y = 0$, so $y(y + 1) = 0$. Since $y \neq 0$, we have $y = -1$. Thus, $z = -i$.

Therefore, the non-zero solutions are $z = i$ and $z = -i$.

Step 6: Calculating $\alpha$ and $\beta$

The sum of the non-zero solutions is $\alpha = i + (-i) = 0$. The product of the non-zero solutions is $\beta = (i)(-i) = -i^2 = -(-1) = 1$.

Step 7: Final Calculation

We need to find $4(\alpha^2 + \beta^2)$. $4(\alpha^2 + \beta^2) = 4(0^2 + 1^2) = 4(0 + 1) = 4(1) = 4$

Common Mistakes & Tips

• Remember that $\sqrt{y^2} = |y|$, not just $y$.
• Pay attention to the condition that we are looking for non-zero solutions.
• Be careful with the signs when dealing with complex conjugates and imaginary numbers.

Summary

By taking the conjugate of the given equation and equating the squared terms, we were able to find that the non-zero solutions are $i$ and $-i$. Then, we calculated the sum and product of the solutions, and finally, we found the value of $4(\alpha^2 + \beta^2)$.

The final answer is $\boxed{4}$, which corresponds to option (A).