Let z be a complex number with non-zero imaginary part. If (2 + 3z + 4z²)/(2 - 3z + 4z²) is a real n

Q: Wait, why do we rewrite the numerator as $(2 - 3z + 4z^2) + 6z$?

This is a classic algebraic technique to simplify rational expressions. By forcing the denominator to appear in the numerator, we can split the fraction into a constant (which is $1$) plus a simpler remainder. This makes the subsequent conjugation process much cleaner and less prone to calculation errors.

Q: Can we just assume $z = x + iy$ and solve it by expanding everything?

You *could*, but it would lead to a massive, messy system of equations that would be very difficult to solve. The beauty of complex number properties like $w = \bar{w}$ and $z\bar{z} = |z|^2$ is that they allow us to solve the problem elegantly without ever needing to expand $z$ into its real and imaginary parts. Always look for the property-based shortcut first!

Q: Why do we reject the factor $(z - \bar{z}) = 0$?

The problem explicitly states that $\text{Im}(z) \neq 0$. If $(z - \bar{z}) = 0$, then $z = \bar{z}$, which implies that $z$ is a purely real number. Since the problem forbids this, that factor cannot be zero, which leaves only the other factor, $2 - 4|z|^2$, to satisfy the equation.

Question

Let $z$ be a complex number with non-zero imaginary part. If $\frac{2 + 3z + 4z^2}{2 - 3z + 4z^2}$ is a real number, then the value of $|z|^2$ is _____________.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, fellow explorers of the mathematical universe! Today, we are going to tackle a problem that might look like a tangled mess of algebra at first glance, but beneath the surface, it hides a beautiful, symmetrical structure.

We are given the expression:

$$\alpha = \frac{2 + 3z + 4z^2}{2 - 3z + 4z^2}$$

We are told that $\alpha$ is a purely real number, with the crucial constraint that $	ext{Im}(z) 
eq 0$. This constraint is our compass; it tells us that $z$ is not just any number, but a complex one living off the real axis.

### The Algebraic Insight

When you see a rational expression like this, your first instinct might be to panic or start expanding. But pause! Look at the numerator and the denominator; they are almost identical, save for the sign of the middle term.

We can perform a simple algebraic maneuver to simplify our lives. We rewrite the numerator as $(2 - 3z + 4z^2) + 6z$. Now, when we divide this by the denominator, the expression becomes:

$$\alpha = 1 + \frac{6z}{2 - 3z + 4z^2}$$

Suddenly, the intimidating fraction has been tamed into something much more manageable.

### The Conjugate Condition

Now, we invoke the power of the complex conjugate. A complex number $w$ is real if and only if $w = \bar{w}$. Since $\alpha$ is real, we must have $\alpha = \bar{\alpha}$.

Substituting our simplified expression, we get:

$$1 + \frac{6z}{2 - 3z + 4z^2} = \overline{\left(1 + \frac{6z}{2 - 3z + 4z^2}ight)}$$

Using the properties of conjugates, where the conjugate of a sum is the sum of the conjugates and the conjugate of a quotient is the quotient of the conjugates, the equation becomes:

$$1 + \frac{6z}{2 - 3z + 4z^2} = 1 + \frac{6\bar{z}}{2 - 3\bar{z} + 4\bar{z}^2}$$

The constant $1$ cancels out beautifully, and dividing by $6$ leaves us with:

$$\frac{z}{2 - 3z + 4z^2} = \frac{\bar{z}}{2 - 3\bar{z} + 4\bar{z}^2}$$

### The Algebraic Dance

Now, we cross-multiply to clear the fractions:

$$z(2 - 3\bar{z} + 4\bar{z}^2) = \bar{z}(2 - 3z + 4z^2)$$

Expanding these terms gives us:

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

Notice the term $-3z\bar{z}$ on both sides? It vanishes! We are left with:

$$2z + 4z\bar{z}^2 = 2\bar{z} + 4z^2\bar{z}$$

Rearranging everything to one side, we get:

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

Factoring out $(z - \bar{z})$, we arrive at:

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

### The Final Revelation

We know that $z\bar{z} = |z|^2$. So our equation is:

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

Since $	ext{Im}(z) 
eq 0$, we know $z 
eq \bar{z}$, meaning $(z - \bar{z}) 
eq 0$. Therefore, we can safely divide by this term, leaving us with:

$$2 - 4|z|^2 = 0$$

Solving for $|z|^2$, we find $4|z|^2 = 2$, which gives us:

**$|z|^2 = 0.5$**

And there it is! Through careful simplification and the application of complex properties, we have arrived at the solution. It is a reminder that in mathematics, the most complex problems often yield to the most elegant insights.

Animated Solution for Mathematics - Complex Numbers: Let $z$ be a complex number with non-zero imaginary part. If $\frac{2 + 3 z + 4 z ^{2}}{2 - 3 z + 4 z ^{2}}$ is a real number, then the value of $∣ z ∣^{2}$ is _____________.

Enter Numerical Value:

Visualized Solution

Given Expression

Simplifying $α$

Condition for Real Number

Substituting $α$

Simplifying the Equation

Cross-Multiplying

Expanding the Terms

Canceling Common Terms

Rearranging the Equation

Grouping Terms

Factoring out $(z - \overset{z}{ˉ})$

Using $z \overset{z}{ˉ} = ∣ z ∣^{2}$

Analyzing the Condition

Rejecting $z = \overset{z}{ˉ}$

Solving for $∣ z ∣^{2}$

The Sigma Insight: Algebraic Operations on Complex Numbers

Analyzing the Setup

The Algebraic Insight

The Conjugate Condition

The Algebraic Dance

The Final Revelation

Similar Questions

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Enter Numerical Value:

Visualized Solution

Given Expression

Simplifying .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } α

Condition for Real Number

Substituting .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } α

Simplifying the Equation

Cross-Multiplying

Expanding the Terms

Canceling Common Terms

Rearranging the Equation

Grouping Terms

Factoring out .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (z−zˉ)

Using .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } zzˉ=∣z∣2

Analyzing the Condition

Rejecting .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } z=zˉ

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∣z∣2