What is the quadratic equation and what does it do?
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An equation is something with an = in it. One thing (or set of things) is the same as another thing.
Usually equations will involve unknowns (things we don’t know but want to find out) and things we do know, and the goal is to use them (and the various mathematical rules we have) to figure out what the unknowns are or could be.
The simplest equations we have are linear equations, they look like:
> ax + b = 0
We have a constant term (b) and an “x” term multiplied by some constant (a) scale factor. We can always solve this by rearranging to get a single answer for “x”:
> x = -b/a
The next simplest kind of equations we get are where we also throw in an x^2 term:
> ax^2 + bx + c = 0
we have our constant, our linear term, and now a quadratic term in that “x^(2).”
Quadratic equations – being one of the most basic forms of equation we can get – crop up all over the place. Particularly with things that change. Anything that changes at a constant rate gives us a linear equation. Anything that changes at a rate that changes at a constant rate gives us a quadratic.
Because they turn up so often, and are relatively simple, it is worth it for us to just memorise the solutions, or solve them in their most general form.
And the neat thing about quadratic equations is that there is a *general solution* to them. Given any constants a, b, and c (with a not being 0) we can find solutions:
> x = -b/2a + sqrt(b^2 – 4ac)/2a
and
> x = -b/2a – sqrt(b^2 – 4ac)/2a
No matter the values of a, b and c, these values will solve our equation.
We can also look at quadratic functions:
> f(x) = ax^2 + bx + c
As functions these things will take different values depending on what we put in as “x.” Here “x” isn’t an *unknown* but a *variable*; it varies, taking a bunch of different values. And for each possible x we put into our function we will get out a specific value for *f*.
Because quadratics turn up so often these are also worth studying in the most general form; we can look at their behaviour for different values of a, b and c. We can look at the patterns they have, the kinds of solutions we get for *f(x) = k* and so on.
Quadratics are neat equations; they provide a certain non-trivial level of complexity, while also being completely solvable in general terms. Which also makes them great for teaching students key mathematical ideas.
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