If a+bi represents the complex number that result from multiplying `3+2i` times `5-i`, what is the value of a?

Options

A. `2`
B. `13`
C. `15`
D. `17`

Answer

To find the value of \( a \) in the complex number \( a + bi \) that results from multiplying \( 3 + 2i \) by \( 5 - i \), we can follow these steps:

### Step 1: Write down the multiplication of the complex numbers
We need to multiply the two complex numbers:
\[ (3 + 2i)(5 - i) \]

### Step 2: Distribute the terms
Using the distributive property (also known as the FOIL method for binomials), we multiply each term in the first complex number by each term in the second complex number:
\[ = 3 \cdot 5 + 3 \cdot (-i) + 2i \cdot 5 + 2i \cdot (-i) \]

### Step 3: Calculate each multiplication
Calculating each term gives us:
\[ = 15 - 3i + 10i - 2i^2 \]

### Step 4: Substitute \( i^2 \) with \(-1\)
We know that \( i^2 = -1 \), so we replace \( -2i^2 \) with \( -2(-1) \):
\[ = 15 - 3i + 10i + 2 \]

### Step 5: Combine like terms
Now, we combine the real parts and the imaginary parts:
\[ = (15 + 2) + (-3i + 10i) \]
\[ = 17 + 7i \]

### Step 6: Identify \( a \) and \( b \)
From the expression \( 17 + 7i \), we can identify:
\[ a = 17 \quad \text{and} \quad b = 7 \]

### Conclusion
Thus, the value of \( a \) is:
\[ \boxed{17} \]