# Quadratic Formula: Where Did It Come From? **URL:** \ **Category:** Factual Questions\ **Created:** [March 27, 2003, 12:33pm UTC](https://boards.straightdope.com/t/quadratic-formula-where-did-it-come-from/164503 "2003-03-27T12:33:52Z")\ **Posts on this page:** 15\ **Page:** 1
**Author:** ![Jinx](https://avatars.discourse-cdn.com/v4/letter/j/c6cbf5/32.png) [@Jinx](https://boards.straightdope.com/u/Jinx)\ **Post date:** [March 27, 2003, 12:33pm UTC](https://boards.straightdope.com/t/quadratic-formula-where-did-it-come-from/164503/1 "2003-03-27T12:33:52Z")
I am not asking the SDopers to derive the quadratic formula! But, I am curious how we were able to derived it. What method from Calculus (I assume) gave us this formula for factoring ALL quadratics? Any math wizards care to explain? I am especially curious where the “-4ac” term comes from…under the radical. But, if it’s too messy to get into, I’ll just accept it. - Jinx ---
**Author:** ![Desmostylus](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@Desmostylus](https://boards.straightdope.com/u/Desmostylus)\ **Post date:** [March 27, 2003, 12:38pm UTC](https://boards.straightdope.com/t/quadratic-formula-where-did-it-come-from/164503/2 "2003-03-27T12:38:00Z")
[Quadratic Equation](http://mathworld.wolfram.com/QuadraticEquation.html) ---
**Author:** ![CalMeacham](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/calmeacham/32/35_2.png) [@CalMeacham](https://boards.straightdope.com/u/CalMeacham)\ **Post date:** [March 27, 2003, 12:40pm UTC](https://boards.straightdope.com/t/quadratic-formula-where-did-it-come-from/164503/3 "2003-03-27T12:40:22Z")
The link gives the method to derive it. I remember learning to derive the quadratic formula 'way back in Algebra I in junior high, and we did it by “completing the square”. It’s straightforward algebra, exact and accurate, and no calculus is involved. ---
**Author:** ![Uncommon\_Sense](https://avatars.discourse-cdn.com/v4/letter/u/f9ae1b/32.png) [@Uncommon\_Sense](https://boards.straightdope.com/u/Uncommon_Sense)\ **Post date:** [March 27, 2003, 12:56pm UTC](https://boards.straightdope.com/t/quadratic-formula-where-did-it-come-from/164503/4 "2003-03-27T12:56:03Z")
> [@](#): > > \*Originally posted by CalMeacham \* > \*\*The link gives the method to derive it. I remember learning to derive the quadratic formula 'way back in Algebra I in junior high, and we did it by “completing the square”. It’s straightforward algebra, exact and accurate, and no calculus is involved. \*\* I remember that too. Our teacher was so insane that she made that proof half of the final that semester. ---
**Author:** ![Jinx](https://avatars.discourse-cdn.com/v4/letter/j/c6cbf5/32.png) [@Jinx](https://boards.straightdope.com/u/Jinx)\ **Post date:** [March 27, 2003, 1:17pm UTC](https://boards.straightdope.com/t/quadratic-formula-where-did-it-come-from/164503/5 "2003-03-27T13:17:20Z")
Wait a second, I’m missing something here (looking at the derivation found at the link given above). What happened to the linear “x” variable in the problem? When I complete the square, and expand it… I end up with (bx/2a) + (bx/2a) = (2b/2a)x = (b/a)x term. So, where did the linear “x” variable go? 😕 Caution: I haven’t had my coffee yet, so feel free to point out the obvious (gently)… 😃 - Jinx ---
**Author:** ![Jinx](https://avatars.discourse-cdn.com/v4/letter/j/c6cbf5/32.png) [@Jinx](https://boards.straightdope.com/u/Jinx)\ **Post date:** [March 27, 2003, 1:18pm UTC](https://boards.straightdope.com/t/quadratic-formula-where-did-it-come-from/164503/6 "2003-03-27T13:18:51Z")
Oh wait…the trick is NOT to expand the left hand side of the equation so we can then take the square root of both sides! A-ha! Still, it seems like we’re cheating a bit to make this work! - Jinx :o ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [March 27, 2003, 2:44pm UTC](https://boards.straightdope.com/t/quadratic-formula-where-did-it-come-from/164503/7 "2003-03-27T14:44:55Z")
> [@](#): > > \*Originally posted by Jinx \* > \*\*Oh wait…the trick is NOT to expand the left hand side of the equation so we can then take the square root of both sides! > A-ha! Still, it seems like we’re cheating a bit to make this work! > > - Jinx :o \*\* As long as the steps are reversible (and they are), there’s no problem. ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [March 27, 2003, 3:08pm UTC](https://boards.straightdope.com/t/quadratic-formula-where-did-it-come-from/164503/8 "2003-03-27T15:08:57Z")
Cheating? Do you mean, it looks like they pulled a trick out of a hat, and it just happens to work? That’s the name of the game, actually. All mathematics is a search for that sort of thing–the right trick. Without really cheating, of course. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [March 27, 2003, 5:37pm UTC](https://boards.straightdope.com/t/quadratic-formula-where-did-it-come-from/164503/9 "2003-03-27T17:37:41Z")
Of course, once you have the quadratic formula, you can verify it without any further “tricks”, just some crank-turning algebra. So if your teacher wants you to “prove” the quadratic formula, the way to start is to take a “guess” as to what it might be and check that “guess”, and by “pure luck” just happen to choose the right form as your first guess. ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [March 27, 2003, 6:13pm UTC](https://boards.straightdope.com/t/quadratic-formula-where-did-it-come-from/164503/10 "2003-03-27T18:13:06Z")
**Desmostylus** ’s link is really just a proof of correctness of the formula. It doesn’t _explain_ at all where it came from. A proof of correctness for such formulas is easy: Plug in the answer and simplify. In this case take (x-first root)(x-second root), multiply thru and simplify. If you take the sequence of formulas and write them in backwards order, you have an alleged “derivation”. But it is completely unhelpful and _shouldn’t be taught_. This is one way to tell how bad a Math teachers is. “Presto here it is” vs. “here’s some _knowledge_ about solving problems.” Here’s how to truly derive a solution: Note that a two term quadratic is easy to solve: 0 = dz[sup]2[/sup]+e z = +|- sqrt(-e/d) It is easy to solve since the parabola is symmetric about the y-axis. A three term quadratic (with non-zero b) isn’t symmetric in y. Let’s do the following: 1. “Push the parabola sideways” until it’s symmetric around y. 2. Solve the easy case. 3. Take back the amount we “pushed” to get our real answer. How much to push? The “center” (minima or maxima) of a parabola is at -b/(2a). We _could_ get this from Calculus, but that’s too advanced for this problem. Note that the curve crosses the y-axis at x=0, at height y=c. Where else is y=c? Solving ax[sup]2[/sup]+bx+c=c is easy since the c’s cancel out. One solution is x=0 of course, the other is x=-b/a. This means the amount to push (where the center is) is half -b/a. So let’s sub: x = z-b/(2a) in the orig. equation, the z terms cancel, allowing us to easily solve for z as above: z = +|- sqrt(b[sup]2[/sup]-4ac)/(2a) Adding back the “push” amount of -b/(2a) gives the quadratic formula. It is a lot better to learn _methods_ of problem solving that mere facts. ---
**Author:** ![Monty](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/monty/32/47_2.png) [@Monty](https://boards.straightdope.com/u/Monty)\ **Post date:** [March 27, 2003, 6:25pm UTC](https://boards.straightdope.com/t/quadratic-formula-where-did-it-come-from/164503/11 "2003-03-27T18:25:22Z")
I’m pretty sure the Quadratic Forumula was discovered using Algebra. My college Algebra teacher advised: “If you forget the formula during the exam, don’t panic! Just derive it.” That scared me so much, I memorized the thing! ---
**Author:** ![King\_Bubba\_the\_Wise](https://avatars.discourse-cdn.com/v4/letter/k/b5a626/32.png) [@King\_Bubba\_the\_Wise](https://boards.straightdope.com/u/King_Bubba_the_Wise)\ **Post date:** [March 27, 2003, 8:12pm UTC](https://boards.straightdope.com/t/quadratic-formula-where-did-it-come-from/164503/12 "2003-03-27T20:12:36Z")
I’m a little hazy but isn’t the quadratic equation just the natural re-statement of ax^2 + bx + c = 0 where x is isolated? In other words, it’s the same damn equation, just re-jiggered around? ---
**Author:** ![li\_l\_Dickie\_Dirtz](https://avatars.discourse-cdn.com/v4/letter/l/2bfe46/32.png) [@li\_l\_Dickie\_Dirtz](https://boards.straightdope.com/u/li_l_Dickie_Dirtz)\ **Post date:** [March 27, 2003, 8:33pm UTC](https://boards.straightdope.com/t/quadratic-formula-where-did-it-come-from/164503/13 "2003-03-27T20:33:44Z")
[Dr. Math](http://mathforum.org/library/drmath/view/53153.html) says some 12th century Italians figured it out. But they probably weren’t the first. ---
**Author:** ![sturmhauke](https://avatars.discourse-cdn.com/v4/letter/s/e47c2d/32.png) [@sturmhauke](https://boards.straightdope.com/u/sturmhauke)\ **Post date:** [March 27, 2003, 8:52pm UTC](https://boards.straightdope.com/t/quadratic-formula-where-did-it-come-from/164503/14 "2003-03-27T20:52:25Z")
> [@](#): > > \*Originally posted by King Bubba the Wise \* > \*\*I’m a little hazy but isn’t the quadratic equation just the natural re-statement of ax^2 + bx + c = 0 where x is isolated? In other words, it’s the same damn equation, just re-jiggered around? \*\* Yes, that’s exactly right. This is true whenever you solve for a variable. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [March 27, 2003, 11:01pm UTC](https://boards.straightdope.com/t/quadratic-formula-where-did-it-come-from/164503/15 "2003-03-27T23:01:18Z")
Actually, a version of the formula was derived by the Babylonians who knew how to find two numbers, given their sum and product. Note that you can divide by a to force a = 1. Now if u and v are the two solutions to z^2 + bx + c = 0, then u + v = b and uv = -c, so every quadratic can be reduced to that problem. This was hundreds of years before the Greek geometers. Later on Diophantus derived it too. The real problem was conceptual: dealing with negaitve numbers, not to imagine complex numbers. For that reason, the equations x^2 + bx + c = 0 was consider different from x^2 + bx = c, x^2 + c = bx, and x^2 = bx + c were all considered different and each had its own solution. In the 15th century, I believe, 3rd and 4th degree analogs of these formulas were found by Italian algebraists. Thus began a search for a 5th degree formula, which ended only in the 1800-30 era with several proofs of impossibility.