If `z=2-3i` show that `z^2=4z+13=0` and hence find the value of `4z^3-3z^2+169.`

Explanation

To solve the problem, we will follow the steps outlined in the video transcript to show that \( z^2 - 4z + 13 = 0 \) and then find the value of \( 4z^3 - 3z^2 + 169 \).

### Step 1: Calculate \( z^2 \)

Given \( z = 2 - 3i \), we first calculate \( z^2 \):

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

Using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \):

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

Since \( i^2 = -1 \):

\[ = 4 - 12i - 9 \]
\[ = -5 - 12i \]

### Step 2: Substitute \( z \) into \( z^2 - 4z + 13 \)

Now, we need to show that \( z^2 - 4z + 13 = 0 \):

\[ z^2 - 4z + 13 = (-5 - 12i) - 4(2 - 3i) + 13 \]

Calculating \( -4(2 - 3i) \):

\[ -4(2 - 3i) = -8 + 12i \]

Now substitute this back into the equation:

\[ (-5 - 12i) + (-8 + 12i) + 13 \]
\[ = -5 - 8 + 13 + (-12i + 12i) \]
\[ = 0 \]

Thus, we have shown that:

\[ z^2 - 4z + 13 = 0 \]

### Step 3: Find the value of \( 4z^3 - 3z^2 + 169 \)

To find \( 4z^3 - 3z^2 + 169 \), we first need to express \( z^3 \) in terms of \( z \) and \( z^2 \).

Using the relation \( z^2 = 4z - 13 \):

\[ z^3 = z \cdot z^2 = z(4z - 13) = 4z^2 - 13z \]

Now substituting \( z^2 \) into the equation:

\[ z^3 = 4(4z - 13) - 13z \]
\[ = 16z - 52 - 13z \]
\[ = 3z - 52 \]

Now substituting \( z^3 \) into \( 4z^3 - 3z^2 + 169 \):

\[ 4z^3 - 3z^2 + 169 = 4(3z - 52) - 3(4z - 13) + 169 \]
\[ = 12z - 208 - 12z + 39 + 169 \]
\[ = -208 + 39 + 169 \]
\[ = 0 \]

### Final Result

Thus, the value of \( 4z^3 - 3z^2 + 169 \) is:

\[ \boxed{0} \]