Exploring the usefulness and (mostly) non-usefulness of the quadratic formula

#### Quadratic Formula; The least useful tool of the topic!

Generally, most students try to memorize the quadratic formula and ignore most of
the other content on quadratics for the simple fact that the quadratic formula
*always* gives the zeros of a quadratic whereas the other methods often are
considerably more work (at least initially when learning them) and don’t always
yield a useful answer. Nonetheless the quadratic formula is, by far, the least
useful thing in this section for the simple fact that it *only works on quadratic
forms*. In this class as well as calculus, we will typically have higher degree
polynomials that we need to deal with, where the quadratic formula doesn’t
apply.

Nonetheless, there are times when factoring is not a viable option. In these cases the
quadratic formula is quite helpful, and thus you should think of this as a niche tool; a
formula that has a **very specific circumstance** in which it can and should be
used.

#### The Quadratic Formula

The quadratic formula is used to find the *zeros* of a quadratic. In particular this
means it should be written *as the full equation*.

We shouldn’t take on faith that this equation works, instead we should show how we
know for certain that this equation works, ie we should *prove* it.

To do this, recall that what we want are the zeros of , which is to say we want the values so that . So we want to solve:

As it happens, we can actually solve this, despite the remarkable level of generality by completing the square!

Notice that the equation makes no assumptions to suggest that the expression is
non-negative. In fact, we can learn a lot thanks to the value of this expression, which
also gets a special name; the *discriminate*.

In general, the discriminate is some sort of ‘distance’ between the vertex and the
zeros of the function. Thus if the discriminate is positive (noting as well that a
quadratic is symmetric across the vertex due to its ‘U’ shape), then it will
have 2 (real-valued) zeros, one on each side of the vertex. If the discriminate
is zero, then the quadratic will have the zeros *at* the vertex, meaning the
there is one root twice. Finally, if the discriminate is negative, this means
that the ‘distance’ between the vertex and zero is imaginary. This can only
happen if *there are no* real-valued zeros. Since we know the roots exist by the
fundamental theorem of algebra, they must be non-real valued (ie complex
roots).

To put it succinctly:

- If there are 2 real-valued zeros and thus 2 real-valued roots.
- If there is 1 real-valued zero and thus 1 real-valued root with multiplicity 2 or a ‘double root’.
- If there are 0 real-valued zeros and thus 2 complex-valued (ie non-real) roots.