Question

How do you simplify `( 3 + i ) ( 3 + i )`?

Answer 1

FOIL and know that `i=sqrt(-1)`, so `i^2=-1`.

Explanation:

If the problem were to simplify `(5+x)(2-x)`, you would get `10+2x-5x-x^2`. This problem uses the same logic, except with complex numbers. (Remember that `i=sqrt(-1)`.)

We must distribute `(3+i)(3+i)`.
We should get: `9+3i+3i+i^2`
Combine like terms: `9+6i+color(red)(i^2)`

Now, it may seem that we have done all we can. However, since `i=sqrt(-1)`, we can say that `color(red)(i^2)=(sqrt(-1))^2=color(red)(-1)`. We can put this back into our simplification.

`9+6i+(color(red)(-1))`

Then, we combine like terms again, leaving us with our final answer:
`color(blue)(8+6i)`