solve 
𝑧
2
=
3
+
4
𝑖
z 
2
 =3+4i for complex numbers

We are tasked with solving the equation \( z^2 = 3 + 4i \), where \( z \) is a complex number. Let's proceed step by step:

### Step 1: Express \( z \) as a complex number
Let \( z = a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit (where \( i^2 = -1 \)).

### Step 2: Square \( z \)
We need to square \( z \):

\[
z^2 = (a + bi)^2 = a^2 + 2abi + (bi)^2
\]
Since \( i^2 = -1 \), this simplifies to:

\[
z^2 = a^2 + 2abi - b^2
\]

Thus, the real part of \( z^2 \) is \( a^2 - b^2 \), and the imaginary part is \( 2ab \). Therefore:

\[
z^2 = (a^2 - b^2) + 2abi
\]

### Step 3: Set \( z^2 \) equal to \( 3 + 4i \)
Now, equate the real and imaginary parts:

\[
a^2 - b^2 = 3 \quad \text{(Real part)}
\]
\[
2ab = 4 \quad \text{(Imaginary part)}
\]

### Step 4: Solve the system of equations
From the imaginary part equation \( 2ab = 4 \), solve for \( ab \):

\[
ab = 2
\]

Now we have two equations:

1. \( a^2 - b^2 = 3 \)
2. \( ab = 2 \)

### Step 5: Solve for \( a \) and \( b \)
Using \( ab = 2 \), we can express \( b \) in terms of \( a \):

\[
b = \frac{2}{a}
\]

Substitute this into the equation \( a^2 - b^2 = 3 \):

\[
a^2 - \left(\frac{2}{a}\right)^2 = 3
\]

Simplify:

\[
a^2 - \frac{4}{a^2} = 3
\]

Multiply through by \( a^2 \) to eliminate the denominator:

\[
a^4 - 4 = 3a^2
\]

Rearrange:

\[
a^4 - 3a^2 - 4 = 0
\]

Let \( x = a^2 \), so the equation becomes:

\[
x^2 - 3x - 4 = 0
\]

### Step 6: Solve the quadratic equation
Solve \( x^2 - 3x - 4 = 0 \) using the quadratic formula:

\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-4)}}{2(1)}
\]
\[
x = \frac{3 \pm \sqrt{9 + 16}}{2} = \frac{3 \pm \sqrt{25}}{2} = \frac{3 \pm 5}{2}
\]

Thus, we get two solutions for \( x \):

\[
x = \frac{3 + 5}{2} = 4 \quad \text{or} \quad x = \frac{3 - 5}{2} = -1
\]

Since \( x = a^2 \) and \( a^2 \geq 0 \), discard the negative solution \( x = -1 \). So, \( a^2 = 4 \), which gives \( a = 2 \) or \( a = -2 \).

### Step 7: Find \( b \)
Since \( ab = 2 \), if \( a = 2 \), then:

\[
b = \frac{2}{2} = 1
\]

If \( a = -2 \), then:

\[
b = \frac{2}{-2} = -1
\]

### Step 8: Final solution
Thus, the two possible solutions for \( z \) are:

\[
z = 2 + i \quad \text{or} \quad z = -2 - i
\]

### Conclusion
The solutions to the equation \( z^2 = 3 + 4i \) are:

\[
z = 2 + i \quad \text{or} \quad z = -2 - i
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you square a complex number?
2. What is the general process for solving quadratic equations with complex numbers?
3. Can you explain the significance of equating real and imaginary parts?
4. What are polar forms of complex numbers, and how can they help in solving equations like this?
5. How can we visualize complex numbers and their operations?

### Tip:
When solving complex number equations, always break the problem into real and imaginary parts for clarity.

This detailed guide walks through solving the complex number equation z^2 = 3 + 4i, covering essential algebraic methods like squaring complex numbers, solving quadratic equations, and using real and imaginary parts. Suitable for advanced high school students (Grades 10-12).

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