Question

How do you simplify `(3 + 4i) (3 + 4i)`?

Answer 1

`-7+24i`

Explanation:

Your first step would be just like if you multiplied together two binomials: factor out each to get a trinomial. If you use FOIL, it's easy: multiply the First terms, the Outer terms, the Inner terms, and the Last terms:

F: `3*3=9`
O: `3*4i=12i`
I: `4i*3=12i`
L: `4i*4i=16i^2`

Sum these together, and you can get your halfway-simplified version:

`16i^2+12i+12i+9=16i^2+24i+9`

We know that `i=sqrt(-1)`, so if we have `i^2`, we're really just saying `(sqrt(-1))^2` which simplifies to `-1`. Knowing this, we can simplify farther yet:

`16*(-1)+24i+9`
`-16+24i+9`
`24i-7`

Since standard imaginary form is `a+bi`, we can swap it around to

`-7+24i` to get our final answer.