Find the fixed points of the bilinear map
𝑤𝑤 = 2𝑖𝑖−6�

To find the fixed points of the bilinear map, we need to solve for the points where \( w = z \). The given bilinear map is:

\[
w = \frac{2i z - 6}{z - 3}
\]

A fixed point occurs when \( w = z \). So we set \( w = z \) in the given bilinear map:

\[
z = \frac{2i z - 6}{z - 3}
\]

Now, we solve this equation for \( z \):

1. **Multiply both sides by \( z - 3 \) to clear the denominator:**

\[
z(z - 3) = 2i z - 6
\]

2. **Expand and rearrange the equation:**

\[
z^2 - 3z = 2i z - 6
\]

\[
z^2 - 3z - 2i z + 6 = 0
\]

3. **Combine like terms:**

\[
z^2 - (3 + 2i)z + 6 = 0
\]

This is a quadratic equation in the form \( az^2 + bz + c = 0 \), where \( a = 1 \), \( b = -(3 + 2i) \), and \( c = 6 \).

4. **Solve for \( z \) using the quadratic formula:**

\[
z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substitute \( a = 1 \), \( b = -(3 + 2i) \), and \( c = 6 \) into the formula:

\[
z = \frac{-(-3 - 2i) \pm \sqrt{(-3 - 2i)^2 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}
\]

\[
z = \frac{3 + 2i \pm \sqrt{(3 + 2i)^2 - 24}}{2}
\]

5. **Calculate \( (3 + 2i)^2 \):**

\[
(3 + 2i)^2 = 9 + 12i + 4i^2 = 9 + 12i - 4 = 5 + 12i
\]

6. **Now substitute back:**

\[
z = \frac{3 + 2i \pm \sqrt{(5 + 12i) - 24}}{2}
\]

\[
z = \frac{3 + 2i \pm \sqrt{-19 + 12i}}{2}
\]

7. **Finding the square root of \( -19 + 12i \) can be done using the polar form. Let \( r = \sqrt{(-19)^2 + 12^2} \) and \( \theta = \tan^{-1}\left(\frac{12}{-19}\right) \). After calculating the square root, use:**

\[
z_1 = \frac{3 + 2i + \sqrt{-19 + 12i}}{2}
\]

\[
z_2 = \frac{3 + 2i - \sqrt{-19 + 12i}}{2}
\]

These \( z_1 \) and \( z_2 \) values are the fixed points of the bilinear map.

### Summary

The fixed points of the bilinear map \( w = \frac{2iz - 6}{z - 3} \) are the solutions to the quadratic equation \( z^2 - (3 + 2i)z + 6 = 0 \). Solving this using the quadratic formula gives the fixed points.

### Do you want details or have any questions?

Here are some related questions to deepen your understanding:

1. What is the geometric interpretation of fixed points in a bilinear map?
2. How do fixed points of a map relate to its critical points?
3. What is the significance of the discriminant in finding fixed points?
4. How do we find fixed points for a nonlinear transformation?
5. What are the applications of fixed points in complex dynamics?

**Tip:** When solving quadratic equations with complex coefficients, keep track of real and imaginary parts separately for accurate calculations.

Explore how to find the fixed points of the bilinear map w = (2iz - 6) / (z - 3) through a detailed quadratic equation solution. Learn about bilinear maps, complex numbers, and quadratic equations in advanced mathematics.

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