That's very cool. To the best of my knowledge the only purpose of learning how to complete the square in high school math is to be able to understand the derivation of the quadratic formula, so if we can use this alternative proof then students don't need to know "completing the square" anymore. Good riddance!
In sum, completing the square is definitely a cool trick (see https://en.wikipedia.org/wiki/File:Completing_the_square.gif ), but maybe we can skip it... or present it as extra/optional material? I'm going to think about dropping it from my books. It will save 5+ pages of suffering for readers, which is a clear win.
Nope, completing the square is primarily used to translate a standard quadratic function into a vertex form quadratic so it can be graphed using transformations instead of guessing.
It is an essential multi-use tool in precalculus. The thing we should dump is the quadratic formula.
Comments
That's very cool. To the best of my knowledge the only purpose of learning how to complete the square in high school math is to be able to understand the derivation of the quadratic formula, so if we can use this alternative proof then students don't need to know "completing the square" anymore. Good riddance!
The only other application of "completing the square" I can think of is the trig substitution in calculus, see https://en.wikipedia.org/wiki/Completing_the_square#Integrat... or worked example here https://www.khanacademy.org/math/ap-calculus-ab/ab-integrati...
In sum, completing the square is definitely a cool trick (see https://en.wikipedia.org/wiki/File:Completing_the_square.gif ), but maybe we can skip it... or present it as extra/optional material? I'm going to think about dropping it from my books. It will save 5+ pages of suffering for readers, which is a clear win.
Nope, completing the square is primarily used to translate a standard quadratic function into a vertex form quadratic so it can be graphed using transformations instead of guessing.
It is an essential multi-use tool in precalculus. The thing we should dump is the quadratic formula.