Classify and check a second-order equation

You will write three small functions for the equation $ay'' + by' + cy = 0$. The first finds the two roots of the auxiliary equation. The second names the case. The third tests a function in the equation.

The idea. The roots of $am^2 + bm + c = 0$ are $\dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. The sign of $b^2 - 4ac$ names the case. A function $y$ solves the equation when $ay'' + by' + cy$ is zero. The first and second derivatives can be estimated from nearby values.

Your tasks.

1. Finish roots(a, b, c). Return the pair of roots, using cmath.sqrt so that complex roots work too.
2. Finish case_of(a, b, c). Return "real" when $b^2 - 4ac > 0$, "equal" when it is within 0.000001 of zero, and "complex" otherwise.
3. Finish residual(y, a, b, c, x). Return $ay'' + by' + cy$ at x, with the slope from (y(x + h) - y(x - h)) / (2 * h) and the second derivative from (y(x + h) - 2 * y(x) + y(x - h)) / h ** 2, where h = H.