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Factoring the Sum of Two Squares

Remember the difference of two squares?

We can factor this guy:

x^2 - 9 = ( x - 3 ) ( x + 3 )

You've probably been told that you couldn't factor this guy...

x^2 + 9
The SUM of two squares.

Well, they fibbed...  You CAN factor this guy...  Just not with real numbers.

We just need one more thing though:

Since   

  i = sqrt( -1 )

Squaring both sides gives

i^2 = -1

So, check this out:

I claim that

x^2 + 9 factors as ( x - 3i ) ( x + 3i )
Just like the difference of two squares...  but with i's after the numbers.