If ` z_(1) = 6 + 9i and z_(2) = 5 + 2i ` then find ` z_(1)/z_(2)`

Answer

To find \( \frac{z_1}{z_2} \) where \( z_1 = 6 + 9i \) and \( z_2 = 5 + 2i \), we will follow these steps:

### Step 1: Write the expression for division
We start with the expression:
\[ \frac{z_1}{z_2} = \frac{6 + 9i}{5 + 2i} \]

### Step 2: Multiply numerator and denominator by the conjugate of the denominator
To eliminate the imaginary part in the denominator, we multiply both the numerator and the denominator by the conjugate of \( z_2 \), which is \( 5 - 2i \):
\[ \frac{6 + 9i}{5 + 2i} \cdot \frac{5 - 2i}{5 - 2i} = \frac{(6 + 9i)(5 - 2i)}{(5 + 2i)(5 - 2i)} \]

### Step 3: Simplify the denominator
The denominator can be simplified using the difference of squares:
\[ (5 + 2i)(5 - 2i) = 5^2 - (2i)^2 = 25 - 4(-1) = 25 + 4 = 29 \]

### Step 4: Expand the numerator
Now, we expand the numerator:
\[ (6 + 9i)(5 - 2i) = 6 \cdot 5 + 6 \cdot (-2i) + 9i \cdot 5 + 9i \cdot (-2i) \]
Calculating each term:
- \( 6 \cdot 5 = 30 \)
- \( 6 \cdot (-2i) = -12i \)
- \( 9i \cdot 5 = 45i \)
- \( 9i \cdot (-2i) = -18i^2 = 18 \) (since \( i^2 = -1 \))

Now, combine these results:
\[ 30 - 12i + 45i + 18 = (30 + 18) + (-12i + 45i) = 48 + 33i \]

### Step 5: Combine the results
Now we can write the complete expression:
\[ \frac{z_1}{z_2} = \frac{48 + 33i}{29} \]

### Step 6: Separate into real and imaginary parts
This can be expressed as:
\[ \frac{48}{29} + \frac{33}{29}i \]

### Final Result
Thus, the result of \( \frac{z_1}{z_2} \) is:
\[ \frac{48}{29} + \frac{33}{29}i \]

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