A simpler form of the quadratic formula

In high school, I had to learn the famous quadratic formula by heart. It states that a quadratic polynomial defined as has roots of the form

I think remembering formulae is a prime example of how math should not be taught. I think the proper way to solve quadratic equations is to complete the square:

First, rewrite the equation to the form

We can rewrite this as

From there, it is easy to take it to

and solve the resulting equation, remembering that has two solutions: and :

So .

If you, for some reason, insist on learning a formula (for example, because it is easier or faster to apply), there is an alternative that I find to be preferable to the classical quadratic formula. By setting and , any quadratic equation of the form can be reduced to the form

which has the solution

and is far more elegant, in my opinion.

In conclusion, I think it’s preferable to not teach the quadratic formula in high school, and instead teach how to complete the square. Not only does this relieve you of the burden of remembering a formula, it is also much more instructive and opens doors to more elaborate algebraic manipulation.