Question

Perform the indicated operation and express the result as a simplified complex number. (3+4i)(3−i)

Answer 1

The question is given below as

`(3+4i)(3-i)`

Concept:

Apply the complex arithmetic rule

`\begin{gathered} (a+bi)(c+di) \\ =(ac-bd)+(ad+bc)i \end{gathered}`

By comparing coefficient with the main question, we will have

`a=3,b=4,c=3,d=-1`

Step 1: Substitute the values in the arithmetic rule

`\begin{gathered} (a+bi)(c+di) \\ (ac-bd)+(ad+bc)i \\ (3*3)-(4*-1)+(3*-1)+(4*3)i \\ =(9+4)+(-3+12)i \\ =13+9i \end{gathered}`

Alternatively,

Use the FOIL method

`\begin{gathered} (3+4i)(3-i) \\ 3(3-i)+4i(3-i) \\ =9-3i+12i-4i^2 \\ =9+9i-4i^2 \\ \text{note :} \\ i^2=-1 \\ =9+9i-4(-1) \\ =9+9i+4 \\ =9+4+9i \\ =13+9i \end{gathered}`

Hence,

The final answer is = 13 + 9i