[HN Gopher] The quadratic formula: a tutorial
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The quadratic formula: a tutorial
Author : jamesfisher
Score : 57 points
Date : 2022-02-24 12:24 UTC (1 days ago)
(HTM) web link (tigyog.app)
(TXT) w3m dump (tigyog.app)
| ogogmad wrote:
| Side remark: The completing the square trick can be easily
| generalised to completing the cube. A few additional
| straightforward manipulations (substitutions) result in something
| called a reduced cubic. You can then express the reduced cubic
| equation as cosh(3 acosh(x)) = C for some constant C (or you can
| use the function "cos" instead of "cosh") and solve for x.
| prezjordan wrote:
| Great format! I liked the halting problem lesson too. Are you
| planning to build a company out of this platform? (It's really
| nice to use)
| jamesfisher wrote:
| That's the plan, and thanks for the compliment! :-) If you try
| writing lessons on the app and have any feedback/requests --
| just let me know, jameshfisher@gmail.com
| evancoop wrote:
| My algebra II teacher sung the formula to the tune of "pop goes
| the weasel."
|
| "X equals negative b / Plus or minus square root / B-squared
| minus four a c / All over 2a"
|
| Decades later, I have never forgotten. Apparently, we should just
| write parodies!
| hrunt wrote:
| My Algebra II teacher used Row, Row, Row Your Boat. She also
| wore a t-shirt with the quadratic equation on it, under her
| sweater. As she finished the song, she pulled off the sweater,
| much to the surprise of the students (the shirt stayed on, so
| things were still safe for school).
|
| I'm still not sure if it was the song or the shock of seeing
| your teacher pull her clothes off. While I can still remember
| the song by heart, I also shudder every time I sing it.
| renewiltord wrote:
| Does anyone else find these mnemonics harder to remember than
| the actual thing itself? I see a lot of people online talk
| about this and something else for sines and cosines and it
| always seems really hard.
|
| I'm not saying I've got some super memory. Just that
| remembering the actual expression is easy. The mnemonic is
| always longer and harder for me to recall.
|
| I can picture the shapeform of the equation itself but cannot
| muster a single of the rhymes. I wonder if it is a difference
| in how we were trained as children.
| causi wrote:
| Ours did it to Frere Jacque.
| isaacimagine wrote:
| Or 'never gonna give you up'.
| moron4hire wrote:
| I got my 4 year old to reliably remember our address by just
| making a simply jingle out of it. The tune is not related to
| anything. The power of music for memory recall is amazing.
| jihadjihad wrote:
| My algebra teacher did it to the tune of the Notre Dame fight
| song, haha. Two decades later and I still remember it!
|
| _x equals the opposite of b
|
| plus or minus the square root of
|
| b-squared minus four a c
|
| all over two times a _
| modal-soul wrote:
| My teacher used the following short story:
|
| The negative boy couldn't decide if he wanted to go to the
| radical party. So he decided to be square, missing out on 4
| awesome chicks. It was all over by 2 AM.
| spacedcowboy wrote:
| My chemistry teacher had a similar acronym for remembering
| which of the anode and cathode where positive and negative.
|
| Penises are not cunts.
| pc86 wrote:
| Same! In fact, I had to sing the entire song in my head to
| remember what the actual formula was.
| jimhefferon wrote:
| I still remember lots of facts about Albania from Cheers:
| https://www.youtube.com/watch?v=-F_tT-q8EF0
| rm445 wrote:
| Nice site, but it makes a meal out of the material. The technique
| of 'completing the square' is about as quick and easy as the
| quadratic formula (indeed, doing it with generic coefficients is
| what gives you the quadratic formula) and only calls on a good
| elementary understanding rather than dredging up a formula from
| the memory banks.
| jamesfisher wrote:
| Yeh, "completing the square" is also how I learned it, but I
| thought this was a nice alternative approach that I hadn't seen
| before.
|
| If you like, you can try writing a lesson using the "completing
| the square" approach! :-) https://tigyog.app/lessons/new
| marginalia_nu wrote:
| I think among the first things they taught us when I took
| physics at university was to forget about the formula, and
| learn to complete the square. Felt they had a fairly low
| opinion of the education we had been given thus far, so we had
| to do it almost as a kata to prepare for the reality of
| actually needing to rely on algebra abilities.
|
| This was followed by years of professors telling us to learn to
| derive all sorts of formulas and identities rather than to
| memorize them.
|
| Learning to derive a formula creates a sort mathematical self-
| sufficiency that you don't get when you memorize a formula.
| analog31 wrote:
| The quadratic formula is a nice toy exercise in learning some of
| the pitfalls of floating point numbers.
| amalcon wrote:
| I have a very distinct memory of the quadratic formula: when my
| high school geometry teacher just assumed we all knew it, but
| most of the class had never seen it before.
|
| This sticks in my memory many, many years later because of the
| hours several of us spent (without being asked to do so)
| memorizing the thing, that being much easier than the version of
| the factorization method that we'd been taught previously. I did
| later learn about other methods (being curious about math, and
| having access to a much better Internet than during high school).
| Still, I don't think I'll ever forget that formula.
| mettamage wrote:
| 3Blue1Brown Episode 1 Lockdown Math, A simpler version of the
| quadratic formula.: https://www.youtube.com/watch?v=MHXO86wKeDY
|
| It's a good lesson, thanks to him I can derive the whole from my
| mind. It did take some time with the second part (finding
| deviation d), but I got there :)
|
| I recommend to watch the video, for people who prefer to read,
| I'll derive the whole thing from my mind (the video makes it
| quite intuitive) and write it down.
|
| ___DEMONSTRATING THE 3BLUE1BROWN AWESOMENESS OUT OF MY OWN
| MIND___
|
| I wasn't doing my best to memorize the intuition when he gave the
| lecture, otherwise it wouldn't have taken me 60 minutes to write
| it down ;-)
|
| It starts with a sketch of a parabola that has 2 roots. Something
| like: x^2-4, see a sketch:
| https://www.desmos.com/calculator/q8mkfjmylc
|
| It then shows that the intuition of the quadratic formula is that
| it's basically the midpoint m plus or minus the deviation d.
|
| So for root left and root right you get:
|
| l, r = m +- d
|
| Sketch a parabola yourself, take the midpoint and draw arrow -d
| and arrow +d to the roots to visually verify that this is true.
|
| In order to make this slightly easier, he suggests to immediately
| take the formula ax^2+bx+c=0 and rescale it to x^2+bx+c=0 and
| work from there. The bx and c are not the same symbols, you
| should see it as b-prime and c-prime in the second equation.
|
| ___GETTING THE MIDPOINT m___
|
| Now you need to figure out what m is.
|
| m is the midpoint, how do we calculate the midpoint
| algebraically?
|
| Well, if you'd average out the roots, then you'd get the
| midpoint, so m = (l+r)/2. "But we don't know the roots!"
|
| That's true, but suppose we would, how would that look like? It
| would look like:
|
| y = (x-l)*(x-r) for roots l and r
|
| Expanding it makes: x^2 -l*x -r*x + l*r --> x^2 - x(l+r) + l*r
|
| Comparing it to x^2 + bx + c this means that: b = -(l+r) [**] and
| c = l*r
|
| Remember that m = (l+r)/2. So we can somehow substitute -(l+r)
| for some version of m. Let's rewrite to see what it is.
|
| m = (l+r)/2, 2*m = l+r, -2*m = -(l+r). -2*m = b (see [**])
|
| therefore b = -2*m. Let's rewrite m in terms of b.
|
| -b = 2*m, -b/2 = m. So m = -b/2.
|
| We have our m in ways we understand! m = -b/2 from our rescaled
| x^2+bx+c=0 equation.
|
| ___GETTING THE DEVIATION d___
|
| How do we get d?
|
| 3Blue1Brown shows a very cool thing here about midpoints
| geometrically and numerically.
|
| 3*5 = 15, 4^2 = 16
|
| 1337 * 1339 = 1338^2 - 1 (checking if correct: 1790243 = 1790244
| - 1, yep)
|
| So the closer the roots are to the midpoint, the closer it's
| looking as if you're multiplying the midpoint with itself (makes
| sense). How does it look like you make the deviation big?
|
| Resuming with 3*5 (my own insights, 3B1B probably showed similar
| things).
|
| l*r = c, m^2 = q (I use c because it's related to the c from
| x^2+bx+c = 0, I use q because I need a letter)
|
| 3*5 = 15, 4^2 = 16
|
| 2*6 = 12, 4^2 = 16
|
| 1*7 = 7, 4^2 = 16 [*]
|
| The deviations are clear to see, they are:
|
| 3,5 = 4 +- 1
|
| 2,6 = 4 +- 2
|
| 1,7 = 4 +- 3
|
| Observation: multiplying the roots l and r gives c. c is somehow
| related to the deviation as we can see that in the first case 16
| - 15 = 1 which is the deviation we want.
|
| Let's try the following, and see how close we get to the
| deviation we want:
|
| d1 = 16 - 15 = 1 --> seems to check out
|
| d2 = 16 - 12 = 4 --> does not check out, we need to square root
| it
|
| d3 = 16 - 7 = 9 --> does not check out, we need to square root it
|
| What did we just calculate?
|
| m^2 - c = d^2
|
| I find it hard to explain why this is the case. 3Blue1Brown might
| have an explanation for it. I simply feel the following "if you
| square a midpoint, and you multiply 2 points equally far away
| from the midpoint in opposite directions, that's kinda like
| weirdly squaring a midpoint, so there must be some relationship
| there. The further those 2 points are, the bigger the deviation
| must be. This deviation cannot be d since we're talking about
| squares here. Ah m^2 - c = d^2! Yea that seems to look alright."
|
| Square rooting it we get:
|
| d1 = 1, from m = 4, l,r = m +- d --> 3,5 = 4 +- 1
|
| d2 = 2, from m = 4, l,r = m +- d --> 2,6 = 4 +- 2
|
| d3 = 3, from m = 4, l,r = m +- d --> 1,7 = 4 +- 3
|
| So let's put it into a formula.
|
| d^2 = m^2 - c --> e.g. 3^2 = 4^2 - 7 (see [*] )
|
| So d = sqrt(m^2 - c)
|
| ___PUTTING IT TOGETHER___
|
| We know the quadratic formula is l, r = m +- d for x^2+bx+c = 0
| which was rescaled from ax^2+bx+c=0 (again the bx+c symbols scale
| with it since I am too lazy to write b-prime and c-prime :D).
|
| We know m. m = -b/2
|
| We know d. d = sqrt(m^2 - c)
|
| So the whole thing for left root l and right root r: l,r = -b/2
| +- sqrt(m^2 - c) for x^2+bx+c=0.
| mhh__ wrote:
| I remember being forced to memorize all kinds of stuff about the
| quadratic formula until one day the lesson was taught by the
| headmaster who used to teach maths but didn't anymore, and he
| said to try derive it, so I did.
|
| That was a big step in simultaneously realizing that mathematics
| could be beautiful but also that mathematics could be ruined by
| schools.
| impossiblefork wrote:
| I myself never learned the quadratic formula, I always just
| used completing the square, i.e. just using the fact that
| x^2+ax, and that you can split the ax in two put the resulting
| rectangles on sides of the square, to make a square with side
| x+a/2, by only adding an a^2/4 which is independent of x. Then
| you know that you can just remove the little a/2 square from
| the big square to relate it to x^2+ax.
|
| I think that was the first case where I showed good taste in
| how to do mathematics.
| xigoi wrote:
| If you complete the square with generic coefficient, you get
| the quadratic formula.
| unnouinceput wrote:
| I must confess somewhere around 45% I got bored, simply clicked a
| lot of wrong answers just to get to the bottom. For me memorizing
| the formula is easier than deriving it using this lesson.
| Nevertheless this is a very good resource, it made it to my
| bookmarks collection
| jamesfisher wrote:
| Thanks for the honesty! What was it that made you drop off?
| E.g. was it too long, or too much tedious algebra, or too much
| basics at the start?
| DaemonAlchemist wrote:
| I dropped off at about 70%. For me, it was a few things 1)
| Too much intro material - The explanation of what a root is
| was too long. By the time I got to the start of the
| derivation, my attention span was already running out, due
| to: 2) Too much tedious algebra - I felt annoyed that I
| needed to do a bunch of basic arithmetic to advance to the
| next section. 3) Complicated derivation - Like another
| commenter mentioned, I think the completing the square method
| would have been shorter and easier than this method.
| shatteredspace wrote:
| I dropped off earlier than 45%, somewhere around 25%. It is
| about deriving the quadratic formula, not about 'what's a
| root'.
|
| What's a root could be a link you click on to give more
| information than a special x value when both side of the
| equation line up (a root is when the graph of a function
| crosses the x-axis).
|
| I like imagery and the clicking to progress to the next
| portion. What would be fantastic in my opinion is to take the
| step-by-step approach of something like WolframAlpha and
| apply it to defined topics with a set of defined examples
| rather than hey provide us an answer and if you get it wrong
| we will take you through a rudiment branch that will take us
| back to the main branch.
| jamesfisher wrote:
| Yeh, I think I tried to pack too much into the lesson.
| "What is a root" could be more like a prereq lesson.
|
| WA's step-by-step feature is amazing dark magic. I'd love
| to replicate that somehow ...
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