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

Nov 14, 2015

FOIL and know that $i = \sqrt{- 1}$, so ${i}^{2} = - 1$.

#### Explanation:

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

We must distribute $\left(3 + i\right) \left(3 + i\right)$.
We should get: $9 + 3 i + 3 i + {i}^{2}$
Combine like terms: $9 + 6 i + \textcolor{red}{{i}^{2}}$

Now, it may seem that we have done all we can. However, since $i = \sqrt{- 1}$, we can say that $\textcolor{red}{{i}^{2}} = {\left(\sqrt{- 1}\right)}^{2} = \textcolor{red}{- 1}$. We can put this back into our simplification.

$9 + 6 i + \left(\textcolor{red}{- 1}\right)$

Then, we combine like terms again, leaving us with our final answer:
$\textcolor{b l u e}{8 + 6 i}$