`(9+2i)(4-3i)-(5-i)(4-3i)`
The expression above is equivalent to which of the following?

Options

A. `7`
B. `14-18i`
C. `25`
D. `16+18i`

Answer

To solve the expression \((9+2i)(4-3i)-(5-i)(4-3i)\), we will break it down into steps.

### Step 1: Expand the first term \((9+2i)(4-3i)\)

Using the distributive property (FOIL method):

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

Calculating each term:

- \(9 \cdot 4 = 36\)
- \(9 \cdot (-3i) = -27i\)
- \(2i \cdot 4 = 8i\)
- \(2i \cdot (-3i) = -6i^2\)

Since \(i^2 = -1\), we have:

\(-6i^2 = -6(-1) = 6\)

Now, combine all the terms:

\[ 36 - 27i + 8i + 6 = 42 - 19i \]

### Step 2: Expand the second term \((5-i)(4-3i)\)

Again, using the distributive property:

\[ (5-i)(4-3i) = 5 \cdot 4 + 5 \cdot (-3i) + (-i) \cdot 4 + (-i) \cdot (-3i) \]

Calculating each term:

- \(5 \cdot 4 = 20\)
- \(5 \cdot (-3i) = -15i\)
- \((-i) \cdot 4 = -4i\)
- \((-i) \cdot (-3i) = 3i^2\)

Again, since \(i^2 = -1\):

\(3i^2 = 3(-1) = -3\)

Now, combine all the terms:

\[ 20 - 15i - 4i - 3 = 17 - 19i \]

### Step 3: Combine the results from Step 1 and Step 2

Now we substitute back into the original expression:

\[ (42 - 19i) - (17 - 19i) \]

Distributing the negative sign:

\[ 42 - 19i - 17 + 19i \]

Now combine like terms:

- Real parts: \(42 - 17 = 25\)
- Imaginary parts: \(-19i + 19i = 0\)

Thus, we have:

\[ 25 + 0i = 25 \]

### Final Answer

The expression \((9+2i)(4-3i)-(5-i)(4-3i)\) is equivalent to \(25\).

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