What is the quadratic equation and what does it do?
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Something a little more deep to mention, since others have done good explanations of what the equations does:
A lot of processes in life follow functions. That just means there is a way to mathematically write what they do as we play time forward. But we often don’t know the function they follow, or know it but it’s really complicated.
So we make a different function that is very close, but a lot simpler. Like smoothing a line a bit so you can draw it with a ruler. One of the ways we do that is called “Taylor approximation”.
How that works doesn’t matter, but a Taylor approximation is a pretty good and pretty natural way to get a function that is good enough. Now why is this relevant?
Say you have some function y=f(x), where we have no idea what f(x) is. A mathematician will do a bit of work and be able to get a number a.
y=a is our “zeroth” approximation. It’s very bad because we are essentially assuming nothing changes.
So we do a bit more work and get two numbers: a and b.
y=ax+b is our “first” approximation. It’s better, but still not very good, since it’s a straight line and not much in nature is straight.
So we do a bit more work and get three numbers: a, b, and c.
y=ax^(2)+bx+c is our “second” approximation, and already pretty good in a small area. It has a curve, a slope, and a height. but we can do better.
And so on…
The reason we use the quadratic formula so much is that in a lot of situations the second Taylor approximation is a very good tradeoff between accuracy and complexity. That means we need a lot of quadratic formulas to solve our smoothed out functions.
(Part of the reason we often stop at the second Taylor approximation is that formulas for solving more than quadratics are **really** hard and often impossible. So it’s all related.)
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