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

Jun 7, 2018

$- 7 + 24 i$

#### 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: $3 \cdot 3 = 9$
O: $3 \cdot 4 i = 12 i$
I: $4 i \cdot 3 = 12 i$
L: $4 i \cdot 4 i = 16 {i}^{2}$

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

$16 {i}^{2} + 12 i + 12 i + 9 = 16 {i}^{2} + 24 i + 9$

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

$16 \cdot \left(- 1\right) + 24 i + 9$
$- 16 + 24 i + 9$
$24 i - 7$

Since standard imaginary form is $a + b i$, we can swap it around to

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