Let z be complex number satisfying |z|³ + 2z² + 4z̄ - 8 = 0, where z̄ denotes the complex conjugate

Q: Why do we take the conjugate of the whole equation? Isn't that just making it more complicated?

It is a classic 'Mirror Strategy'. By creating a twin equation, we can subtract them to cancel out the 'messy' terms like $|z|^3$ and the constant. It turns a non-linear mess into a factorable polynomial. It is not adding complexity; it is revealing the underlying structure.

Q: Why can we just ignore the factor $(z - \bar{z})$? What if it is zero?

Great catch! We do not ignore it; we analyze it. The problem explicitly states $\text{Im}(z) \neq 0$. If $z - \bar{z} = 0$, then $z = \bar{z}$, which means $z$ is purely real. Since the problem forbids this, we know for a fact that $z - \bar{z} \neq 0$, allowing us to safely divide by it.

Q: How do I know when to substitute $z = x + iy$?

Once you have simplified the equation to a condition on $z + \bar{z}$, you have the real part. That is the perfect moment to introduce $z = x + iy$. It bridges the gap between complex algebra and simple real-variable algebra, allowing you to solve for the imaginary part $y$.

Question

Let $z$ be complex number satisfying $|z|^3 + 2z^2 + 4\bar{z} - 8 = 0$, where $\bar{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero. Match each entry in List-I to the correct entries in List-II.

List-I:

P. (P) $|z|^2$ is equal to

Q. (Q) $|z - \bar{z}|^2$ is equal to

R. (R) $|z|^2 + |z - \bar{z}|^2$ is equal to

S. (S) $|z + 1|^2$ is equal to

List-II:

1. (1) 12

2. (2) 4

3. (3) 8

4. (4) 10

5. (5) 7

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are tasked with solving the equation $|z|^3 + 2z^2 + 4\bar{z} - 8 = 0$ under the constraint that $	ext{Im}(z) 
eq 0$.

While substituting $z = x + iy$ is a standard approach, it often leads to cumbersome algebraic expressions. Instead, we will utilize the symmetry of the complex plane to simplify the problem.

### The Mirror Strategy

The presence of both $z$ and $\bar{z}$ suggests we should examine the conjugate of the entire equation. Taking the conjugate of both sides, we obtain:

$$\overline{|z|^3 + 2z^2 + 4\bar{z} - 8} = \bar{0}$$

Since $|z|$ is a real number and the conjugate of a sum is the sum of conjugates, this simplifies to:

$$|z|^3 + 2\bar{z}^2 + 4z - 8 = 0$$

### The Algebraic Cleansing

We now have two equations:
1) $|z|^3 + 2z^2 + 4\bar{z} - 8 = 0$
2) $|z|^3 + 2\bar{z}^2 + 4z - 8 = 0$

Subtracting the second equation from the first eliminates the $|z|^3$ and the constant term:

$$2(z^2 - \bar{z}^2) + 4(\bar{z} - z) = 0$$

Using the difference of squares identity $z^2 - \bar{z}^2 = (z - \bar{z})(z + \bar{z})$, we factor the expression:

$$2(z - \bar{z})(z + \bar{z}) - 4(z - \bar{z}) = 0$$

$$2(z - \bar{z})(z + \bar{z} - 2) = 0$$

### Applying Constraints

Given the constraint $	ext{Im}(z) 
eq 0$, we know that $z 
eq \bar{z}$, which implies $(z - \bar{z}) 
eq 0$.

We can safely divide by $2(z - \bar{z})$ to arrive at the simplified condition:

$$z + \bar{z} = 2$$

Since $z + \bar{z} = 2	ext{Re}(z)$, we conclude that the real part of $z$ is $x = 1$.

### The Final Reveal

Substituting $z = 1 + iy$ into the original equation $|z|^3 + 2z^2 + 4\bar{z} - 8 = 0$, where $|z| = \sqrt{1 + y^2}$:

$$(1 + y^2)^{3/2} + 2(1 + iy)^2 + 4(1 - iy) - 8 = 0$$

$$(1 + y^2)^{3/2} + 2(1 - y^2 + 2iy) + 4 - 4iy - 8 = 0$$

$$(1 + y^2)^{3/2} + 2 - 2y^2 + 4iy + 4 - 4iy - 8 = 0$$

$$(1 + y^2)^{3/2} - 2y^2 - 2 = 0$$

Let $u = 1 + y^2$. Then $y^2 = u - 1$, and the equation becomes $u^{3/2} - 2(u - 1) - 2 = 0$, which simplifies to $u^{3/2} = 2u$. Since $u 
eq 0$, we have $u^{1/2} = 2$, so $u = 4$.

Thus, $1 + y^2 = 4$, which gives $y^2 = 3$, or $y = \pm \sqrt{3}$. The solutions are **$z = 1 \pm i\sqrt{3}$**.

List-I

List-II

Select Matching Pairs:

Visualized Solution

Analyzing the Equation

The Conjugate Strategy

The Conjugated Equation

Subtracting the Equations

Factorizing the Difference

Applying the Imaginary Condition

Finding the Real Part

Substituting back to Original

Expanding and Simplifying

Solving for y

The Complex Number z

Evaluating (P)

Evaluating (Q)

Evaluating (S)

Final Match

The Sigma Insight: Conjugate and Modulus

Analyzing the Setup

The Mirror Strategy

The Algebraic Cleansing

Applying Constraints

The Final Reveal

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