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$Ax^2+Bx+C=0 $ is mentioned as standard form of quadratic equation in every textbook or encyclopaedia, but what's so special about it that its called standard form of quadratic equation. Also, I am aware that standard form of quadratic “Function” is different.

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    $\begingroup$ Several books in the 2nd half of the 1800s used the term "standard form" for this way of writing quadratic equations, but I would expect other terms were also used. As to what's so special, my reaction is what other ways of writing have much use, besides the "completing the square form" (probably known under many different terms)? You might be interested in the historical distinctions between "pure" quadratics and "affected" quadratics. $\endgroup$ Jul 20, 2022 at 16:21
  • $\begingroup$ Funnily, in the class I am teaching right now, I am supposed to call "standard form" the "completed square" form which @DaveLRenfro alludes to. I kind of refuse to do so and call that one the "vertex form" instead, which I believe is pretty standard too, ha ha. $\endgroup$ Jul 23, 2022 at 15:25
  • $\begingroup$ Note that there is a very "standard" way to write polynomials, and for degree 2 it boils down to this. (For degree 1 it would be what is commonly called the "slope-intercept form", for good reasons.) --- Quite generally, I suspect terminology like this is not decided by some central authority with supreme reason, but by half-conscious consensus of textbook publishers and authors and teachers. $\endgroup$ Jul 23, 2022 at 15:36

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Afraid I can't be completely authoritative on this, I'm just throwing it in there, so to speak, but I recall reading a 19th-century text once which covered the subject of quadratics, where the "standard" form was given as $ax^2 + 2 b x + c = 0$.

This of course results in the discriminant having the slightly simpler form $b^2 - a c$, which indeed does make the quadratic formula simpler (unless my skills have completely deserted me): $$x = \dfrac {-b \pm \sqrt {b^2 - ac} } a$$ hence eliminating the need for all those pesky numbers.

Obviously sometime in the following decades the standard way of looking at this subject changed in focus, and simplicity of the quadratic form was placed at a higher priority than simplicity of the resulting formula for its evaluation, but I am afraid I have not performed the research to determine when or how that happened.

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    $\begingroup$ In Germany, it is taught as $x^2 + p x + q = 0$, giving the solution $$x_{1,2} = -\frac{p}{2} \pm \sqrt{(\frac{p}{2})^2 - q}$$ $\endgroup$ Jul 24, 2022 at 15:09
  • $\begingroup$ This is great because it requires $A \neq 0$, which is also a requirement for being a quadratic function at the same time. $\endgroup$ Aug 10, 2022 at 21:10

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