Find all complex numbers z such that z^2=8−6i, and give your answer in the form a+bi. Use the square root symbol '√' where needed to give an exact value for your answer.


To find all complex numbers \( z \) such that \( z^2 = 8 - 6i \), we will solve this by expressing the right-hand side in polar form and applying the square root to obtain \( z \).

### Step 1: Write \( 8 - 6i \) in polar form.

1. **Magnitude**: The magnitude \( r \) of \( 8 - 6i \) is given by:

   \[
   r = |8 - 6i| = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10
   \]

2. **Argument**: The argument \( \theta \) is the angle the complex number makes with the positive real axis, which is:

   \[
   \theta = \tan^{-1} \left( \frac{-6}{8} \right) = \tan^{-1} \left( -\frac{3}{4} \right)
   \]

   Using known values, we get:

   \[
   \theta \approx -0.6435 \text{ radians} \quad \text{(approximately)}
   \]

Thus, the polar form of \( 8 - 6i \) is:

\[
8 - 6i = 10 \text{cis}(-0.6435) \quad \text{where} \ \text{cis}(\theta) = \cos(\theta) + i\sin(\theta)
\]

### Step 2: Solve for \( z \) such that \( z^2 = 8 - 6i \).

To solve \( z^2 = 10 \text{cis}(-0.6435) \), we take the square root of both sides. This gives two solutions:

\[
z = \sqrt{10} \text{cis} \left( \frac{-0.6435 + 2k\pi}{2} \right) \quad \text{for} \ k = 0, 1
\]

The two solutions correspond to the two possible values of \( k \).

### Step 3: Find the two solutions.

For \( k = 0 \):

\[
z_1 = \sqrt{10} \text{cis} \left( \frac{-0.6435}{2} \right) = \sqrt{10} \text{cis}(-0.3218)
\]

For \( k = 1 \):

\[
z_2 = \sqrt{10} \text{cis} \left( \frac{-0.6435 + 2\pi}{2} \right) = \sqrt{10} \text{cis}(2.8198)
\]

### Step 4: Convert back to rectangular form.

Now, convert the polar form back into rectangular form:

1. **For \( z_1 \)**:

\[
z_1 = \sqrt{10} (\cos(-0.3218) + i\sin(-0.3218)) = \sqrt{10}(0.9499 - 0.3127i)
\]

Thus:

\[
z_1 \approx 3.0 - 1.0i
\]

2. **For \( z_2 \)**:

\[
z_2 = \sqrt{10} (\cos(2.8198) + i\sin(2.8198)) = \sqrt{10}(-0.9499 + 0.3127i)
\]

Thus:

\[
z_2 \approx -3.0 + 1.0i
\]

### Final Answer:

The two solutions are approximately:

\[
z_1 \approx 3.0 - 1.0i \quad \text{and} \quad z_2 \approx -3.0 + 1.0i
\]

Would you like more details or have any questions?

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**Related questions:**
1. How do we convert a complex number to polar form?
2. What is De Moivre's Theorem and how is it used?
3. How can we determine the argument of a complex number?
4. What is the geometric interpretation of taking the square root of a complex number?
5. How can we verify the solution by squaring \( z \)?

**Tip:** When finding square roots of complex numbers, it's often helpful to convert them into polar form for easier manipulation.

Find all complex numbers z such that z^2=8−6i, and give your answer in the form a+bi. Use the square root symbol '√' where needed to give an exact value for your answer.

Learn how to solve for all complex numbers z such that z² = 8 - 6i. This problem covers key concepts in complex numbers, polar coordinates, and applying De Moivre's Theorem. Step-by-step solution includes converting to polar form and finding roots. Suitable for advanced high school and college students.

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