solve 𝑧 2

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.