High School: Number and Quantity
The Complex Number System HSN-CN.C.8
8. Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i).
Students should know that complex numbers are everywhere. They're in parabolas and quadratics, but they can be found in polynomial identities as well, even in places where you'd never expect them to be. When we say everywhere, we mean it.
For instance, students should learn that it's possible to factor x2 + 4. They'll freak when you tell them. Give them a glass of water and a fluffy pillow and they'll be fine.
All they have to do is rewrite x2 + 4 as x2 – (-4). Now, as long as they're allowed to use imaginary numbers (and why wouldn't they be?) factoring the difference of perfect squares shouldn't be an issue. We should end up with (x + 2i)(x – 2i).
Students should also know that you aren't doing this to torture them. (Well, maybe a little.) They should know that we use factoring to help solve quadratic equations. That means we can set these equations to zero and solve rather than use the lengthy quadratic formula, which they're probably sick of already.
So it stands to reason that the solutions to the equation x2 + 4 = 0 are 2i and -2i.
A nice, pretty little quadratic equation with real coefficients and then all of a sudden, out of nowhere—imaginary roots. Gotta love math; it keeps you guessing. Let's just hope your students feel the same.