We need to solve the equation:
$\bar{z} = \frac{2}{z} + i$ where $\bar{z}$ represents the complex conjugate of $z$.

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Step 1: Represent the complex number $z$

Let $z = x + yi$, where $x, y \in \mathbb{R}$.
Its complex conjugate is: $\bar{z} = x - yi.$ We substitute these into the equation.

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Step 2: Substitute and rewrite the equation

The equation becomes: $x - yi = \frac{2}{x + yi} + i.$ To simplify, multiply the numerator and denominator on the right-hand side by the complex conjugate of the denominator: $\frac{2}{x + yi} = \frac{2(x - yi)}{(x + yi)(x - yi)} = \frac{2(x - yi)}{x^2 + y^2}.$ Thus, the equation becomes: $x - yi = \frac{2x}{x^2 + y^2} - \frac{2y}{x^2 + y^2}i + i.$

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Step 3: Separate real and imaginary parts

The left-hand side is: $x - yi.$ The right-hand side is: $\left(\frac{2x}{x^2 + y^2}\right) + \left(i - \frac{2y}{x^2 + y^2}i\right).$ Simplify the imaginary part: $i - \frac{2y}{x^2 + y^2}i = \left(1 - \frac{2y}{x^2 + y^2}\right)i.$ So the equation becomes: $x - yi = \frac{2x}{x^2 + y^2} + \left(1 - \frac{2y}{x^2 + y^2}\right)i.$

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Step 4: Equate real and imaginary parts

For the real parts: $x = \frac{2x}{x^2 + y^2}.$ For the imaginary parts: $-y = 1 - \frac{2y}{x^2 + y^2}.$

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Step 5: Solve the equations

Real part equation:

$x(x^2 + y^2) = 2x.$ If $x \neq 0$, we can divide both sides by $x$: x^2 + y^2 = 2. \tag{1}

Imaginary part equation:

$-y = 1 - \frac{2y}{x^2 + y^2}.$ Using $x^2 + y^2 = 2$ from Equation (1): $-y = 1 - \frac{2y}{2}.$ $-y = 1 - y.$ This implies: $0 = 1,$ which is a contradiction unless $y = 0$.

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Step 6: Conclusion

Since $y = 0$, the complex number $z$ must be purely real. From Equation (1): $x^2 + 0^2 = 2 \quad \Rightarrow \quad x^2 = 2 \quad \Rightarrow \quad x = \pm\sqrt{2}.$

Thus, the solutions are: $z = \pm\sqrt{2}.$

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Final Answer

The complex numbers $z$ that satisfy the equation are: $z = \sqrt{2} \quad \text{and} \quad z = -\sqrt{2}.$

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Do you want further details or have questions?

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Related Questions:

1. What is the definition of a complex conjugate?
2. How do you simplify fractions with complex numbers?
3. What is the geometric interpretation of the solutions $z = \sqrt{2}$ and $z = -\sqrt{2}$?
4. Can this method apply to other equations involving complex conjugates?
5. How do we multiply and divide complex numbers?

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Tip: Always equate the real and imaginary parts separately when working with complex equations.