Complex Roots

When working with second order linear differential equations one can use complex roots to find a solution.

Given a 2nd order differential equation:

One can find it's roots by replacing the derivative order with powers instead. The resulting equation is called the characteristic equation, we will be replacing with as it matches our notation for fundamental sets:

We now can find the roots using the quadratic formula or the foil method. These roots are components of our general solution:

See textbook page 136 - equation 10 to continue #incomplete expand

See 1-Second Order Scalar DE (Sect 3-1 to 3-5) and Elementary Differential Equations - Kohler & Johnson - Second Edition - Pg. 122 for source and more information.

I don't know why this is the case, perhaps it has something to do with complex numbers hence the title?