Proving the pq formula for solving quadratic equations
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- čas přidán 28. 02. 2024
- The pq formula is a nice formula for solving the quadratic equation in the form of x^2+px+q=0. US school curriculum usually does not teach the pq formula for solving the quadratic equation as x^2+px+q=0. We often learn the classic quadratic formula for ax^2+bx+c=0 in the US is a very nice formula whenever we cannot solve the quadratic equation by factoring. Subscribe to @bprpmathbasics for more algebra tutorials.
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Helping niece. Getting very high number: x^2+6x-432=0 by factoring and the classic quadratic formula
czcams.com/video/F0g0EuiQ-xo/video.html
Professor can you help me with this i have to write this number in the most simple way possible
(2n)!÷1×3×5×7×.............×(2n-1)
This looks like a double!! For 2n-1 on the bottom
Anyway, 2n! Is 2n (2n-1) (2n-2)... (3) (2) (1)
The odds get cancelled by the bottom, we are left with
2n(2n-2)(2n-4)...(4)(2)
Which is 2^n(n+(n-1)+...+2+1)*
And the summation evaluates to n(n+1)/2
Yield to [2^(n-1)][n(n+1)]
Thanks for this interesting question!
*Edit: sorry it should be 2^n(n!). The example led to correct answer since 3! Is also 3+2+1 which is 6.
So e.g. If n is 3 then 6!/1*3*5
=(2^2)(3*4)
Which is indeed 720/15 which is 48
Hope it helps @@makakh2620
In sweden we learn the pq formula, we never learn the quadratic, if there is a, b and c we can make p=b/a and q=c/a
Well, we learnt that there is one with a, b and c but didn't use it
..both are tought...
but since abc is derived from pq as u already mentioned...
..if p/2 or b/2 is a whole number pq is almost easier to do in ur head...
.. and in sweden thay always teach how to derive the formulas...
...its good to know both .. and b/2a ofc... wich makes most quadratics a head calculation...
if I am understanding this correctly then the way I learned it is if you have the quadratic equation y = ax²+bx+c. You rewrite the equation so you have
y=a(x-h)² + k. h is found by 1/2(-b/a) then k is found by -a(h²) +C . It just seems like he is using the letters p and q instead of h and k. After that when you want to solve for x you use the formula x = h + √(-k/a) and x = h - √(-k/a). In both equations the a is also under the square root.
I've never been tought the quadratic formula, just the pq formula.
Back in the late 1940s in our old one-room school house the teacher still taught abc's not pqr's. I'm not sure when all that changed.
same for me. I had to learn the quadratic formula on my own, when I started at university
@@roger7341 I'm from Germany :)
Here in Germany, it depends on the region ("Bundesland"); some teach pq, some teach abc.
What country?
Huh, I don't think "completing the square" has ever been visualized to me before. The phrase makes a lot more sense with a picture.
You may want to look at his video explaining how to obtain the quadratic formula by completing the square czcams.com/video/AD58TWGIcsQ/video.html
Welcome to 500 BC
Well now’s the time!!
In Germany it’s the opposite, we only learn the pq-formula and not the quadric formula.
It depends on the school, I learned both
Hängt vom Bundesland ab. Z. B. in Bayern ist die abc-Formel allgemein üblich.
@@bjornfeuerbacher5514mal wieder typisch bayern
@@NotBroihon Gymnasien in BaWü meines Wissens auch, Realschulen dort allerdings nicht...
@euerbacher5514 Faszinierend, also hier in NRW habe ich die a-b-c Formel NOCH NIE genutzt oder gelehrt gesehen. Ich weiß sie stand in der Formelsammlung, aber keiner meiner Lehrer, Professoren, Nachhilfeschüler, etc hat jemals darüber geredet.
I first came across this when programming real-time simulations and such. If you set a temporary variable to 'p/2' and such, you can get the solution in fewer mathematical operations, which was faster computationally. But where I learned it, we just started with the quadratic formula and 'brought the '2a' inside the radical' with some algebra. Something like:
midpoint = b/(2*a)
span = sqrt(midpoint*midpoint - c/a)
x1 = midpoint + span
x2 = midpoint - span
(midpoint*midpoint is much faster to compute than midpoint**2 back when we were using FORTRAN 66 lol)
Yes, this is how I do it to when programming too (because speed is very important in my field), though when visualizing it using Desmos or other plotting tools I somehow go back to the quadratic formula as that was how I was taught in middle school. Old habits die hard, they say.
Neat, I think Grant did that same visual proof for the abc quadratic formula during his quarantine math series. It was honestly mind blowing.
or you could do the trick one of my math teachers once taught me, that I teach to my taught me, and that's to divide through by "a" so that your "a" value = 1, then the quadratic formula becomes a lot less "messy". And that's basically the pq formula, except you don't bring the 1/2 into the root.
Quadratic formula examples: czcams.com/video/rLMm21LMIOI/video.html
As a is often 1, if not I simplified the abc formula as B=b/a and C=c/a instead. Then you know where p and q came from.
Another option is to write b/2a as average x (x with a dash above it).
Using "completing the square" was how my teacher taught us the method of solving quadratic equations. Last lesson was then how to derive the pq-formula using this method. Thank you for reminding me how long ago this was.
The PQ formula is just a trivial variation on the quadratic formula. Neither is really better or much faster than the other.
In fact, it increases the chance that you need to deal with non-integer values, thanks to the p/2 that gets squared.
pq formula is actually less common to use for solving quadratic equations
i like pq formula more though, it looks more cool and less complicated to my nucleus brain
Man... I wish I had your videos back when I was in high school.
been only taught the pq-formula and used it ever since. can do it in my head if the numbers are nice
Well, nobody taught me that in school, but what they taught me instead is that when a in ax^2 + bx + c is equal to 1, instead of formulas you can just solve the following system:
x1+x2 = -b
x1*x2 = c
Most times it's much easier then formulas.
equal to 0, not 1!
@@hamgification @F1r1at states that this works as long as a=1, not ax2 +bx + c =1
The vertical axis of symmetry for the parabola at x= -p/2 only depends on p which makes sense. Changing q just moves the parabola vertically.
I never learned anything but the pq-formula (in Germany), but they now also teach the abc-form in junior high school. Nobody uses it in high school, since all problems are adopted to the pq-formula anyway (to avoid inconvenient fractions and such).
Since the (abc) quadratic formula is also derived from completing the square, both formula should the same with the caveat that everything is divided by the exponent before the X^2 part.
And indeed with a bit of manipulation you can get the PQ formula from the (abc) quadratic formula.
One thing -P/2 is the average of the 2 roots, and q is the product of the roots.
Po-Shen Loh style
It is a really complicated formula that was a headache in my middle school, though TBF from time to time it is very useful. But only in certain conditions. Personally always preferred quadratic formula.
The truth is kids learn their abc's before their pqr's, so abc just seems more natural than pqr.
@@roger7341 Weirdly we learned pq first. But I somehow didn't get it, so abc was easier for me.
Huh? Why do you call that formula really complicated? The usual quadratic formula is more complicated.
@@bjornfeuerbacher5514 It is more complex, but it is more generally useful. Plus unlike with a,b,c I never could memorize which is p and which is q.
@@jannegrey593 Just as in the alphabet: first comes p, then q. Ok, and you have to memorize that there is no extra coefficient in front of the x².
The p-q formula that I've seen solves x² + 2px + q = 0, giving the even simpler formula x = −p ± √(p²−q).
In India
I am learn 1.
Factor method ( middle term split)
2. Complete the square method
3 yeah!
Quadratic formula(same like 2.)
here in Nederland the same three in the same order .
1. trying to split b in two terms whose product gets a·c . then factorize . it was called the ‘most elegant way’ .
2. completing the square , we called it ‘splitting off the square’ if this is English ("kwadraat afsplitsen") , namely splitting the c in the smart way. would work always and ever, especially gracious when b/a was even .
3. the thing we also called the abc-formula . but never, never as just a trick , no, the teacher showed where it came from, the step by step proof, by "splitting off the square", several times on three different days in one week .
later on, after Pythagoras, he would add: if say, forty years from now you can only remember Pythagoras' theorema and the abc-formula, and happen to be somehow interested in its derivation , then we can say you've really learnt some mathematica on our school .
With the second method being called the Shri Dharacharya method
@@CheckmateCarnage what we called it's not really matter
How we do it's matter
@@technicallightingfriend4247 yes dude, but I was just saying it was an Indian who first formulated this
@@CheckmateCarnageits Shri Dharacharaya not dinacharya
We in Germany were taught both formulas. However most people (including my math teacher) only ended up using the pq formula even though we were allowed to use the "normal" formula. I personally like the normal one better so that is why I used it in basically every text/exam and I never got in trouble.
Die pq-Formel ist einfacher, wenn der Leitkoeffizient bereits 1 ist, also salopp gesagt, wenn vor dem x² kein Faktor steht.
@@pi_xi Zeig mir ein Beispiel, bei dem die pq-Formel signifikant einfacher ist als die Mitternachtsformel. Ich glaube es gibt genug Beispiele, wo die Mitternachtsformel einfacher ist.
Aus meiner Sicht lohnt es sich kaum, sich die pq-Formel wirklich auswendig zu merken, wenn man die Mitternachtsformel beherrscht. Aber steht ja jedem frei.
@@pi_xi Hier sind ja echt nur deutsche unterwegs :D
In Germany pq Formel is standard
nope. I learned the standard one. Can't remember if pq was even mentioned.
The problem: pq only works fine if b and c are integer multiples of a (or a=1, of course). In quite a few cases you get fractions and then have to calculate with fractions under the square root.
Depends on the federal state. In NRW I learned both formulas but as the leading coefficient is 1 in most cases, the pq formula is easier.
However, I know people from Bavaria, who only learned the abc formula. I also only learned long polynomial division and did not know the Horner's method until I met people from other federal states later.
Nope..
@@pi_xi Even if the leading coefficient is 1, it is not true, that in most cases the pq formula is easier... and if it is easy with pq, it is usually easy with the abc formula as well. But if it is easy with the abc formula, it is not necessarily easy with pq.
@@nikolausmoll9201 If the leading coefficient is 1, the equation is already in the form x²+bx+c = 0 (a is 1 and can be omitted), which is equivalent to x²+px+q=0.
In India we learn both the methods and the pq formula is just known as completing the square method
In both formulars you are completing the square.
This is the formular I learned in school over 50 years ago. We had to memorize this phrase (in Danish):
"x is equal to half of the coefficient of x with the opposite sign, plus or minus the square root of: this quantity squared followed by the last term of the equation with the opposite sign."
In india, we learn both the quadratic formula and completing the square, but not the 'completing the square formula' as shown in the thumbnail. Pretty cool to see it extended like this
in Romania we learn it with delta or smth like that like ∆=b²-4ac and then the rest
That is called the Discriminant.
Fantastic
This pq method is the same as Dr. Po-Shen Loh's method, in which I named it "Baby Vieta Method" or Master Po Method.
wow, really cool formula, in Poland we use standard quadratic formula, but I think I might use this pq version in some cases.
There's also the other form 2c/(-b +- sqrt(b^2 - 4ac)) and the roots it gives are swapped wrt the plus/minus sign compared to the standard form. IIRC it's helpful when solving quadratics numerically when |b|~ sqrt(b^2 - 4ac) as the root found using -b+sign(b) sqrt(b^2 - 4ac) may experience a significant loss of accuracy. But as the roots are swapped wrt the plus/minus, which root this affects is different for each formula. So by using using each form to compute one of the two roots, you can avoid this accuracy issue.
And in the real world, many quadratic equations are only slightly quadratic. We don’t want something which becomes nonsense in the limit as a->0.
What is this blue pen that you are using??
So could you do the same if there was a coefficient “a” and then would you just do divide “p” and “q” by “a”?
If you do this, you may get a fraction over “a”.
description: The pq formula is a nice formula for solving the quadratic equation in the form of x^2+px+q=0. US school curriculum usually does not teach the pq formula for solving the quadratic equation in the form of x^2+px+q=0. We often just learn the classic quadratic formula for ax^2+bx-c-0 in the US is a very nice formula whenever we cannot solve the quadratic equation by factoring.
error: ax^2+bx-c-0
Thanks for letting me know. I will fix it now.
@@bprpmathbasics you’re welcome
I suggest the m-n-formula (which I made up myself ;) ).
For the equation 0.5x² + mx + n = 0, the solutions are - m +- sqrt(m² - 2n).
Looks simpler and is easier to memorize than both the a-b-c formula and the p-q formula. You only have to keep in mind that the formula only works when the coefficient of x² is 0.5.
😂😂😂😂
I love it!
And you can always find a constant to multiply through so you get your .5 coefficient for the x^2 term.
If you don't want the coefficient of x² to be a fraction you can start from
x² + 2mx + n = 0
which has the solutions
x₁,₂ = −m ± √(m² − n)
I've seen similar approaches in (old) algebra books, so the idea is certainly not new.
@@NadiehFan Yeah, when I saw the p/2 and (p/2)², this was my first thought: just absorb the 2. Though I'd choose to write it as x² - 2px + q = 0 having solutions x = p ± √(p² - q) to avoid the leading minus sign.
In Germany we actually learn both, depending on the teacher we get. Some teach the quadratic equation and some the pq-formula, then when we get new teachers every like 4 years they might prefer something else...
So... It's just the quadratic formula with a set to 1, then simplified a bit by moving the /2 part inside some terms?
yap it is, and that's why I see no need for anyone who knows the quadratic formula, that he sould learn the pq formula as well. Sure, anyone can do it.
In Germany, I've learned both formulas, but for general cases, you really just need the pq version! And it's easier! If you have an x² coefficient, just divide the whole equation, so it becomes 1...
BTW, I personally use a third version: x_1/2 = - b/2a +/- sqrt( (b/2a)² - c/a ). Just a question of taste!
With d = -b/2a and e = -c/a, this is just x_1/2 = d + sqrt(d² + e). If a=1, it's just d = -b/2 and e = -c. Easy.
I can also use Δ = d² + e as discriminant, because 4a² is always positive > 0 (a != 0).
I thought at first it was the cubic formula simplified which uses p and q
Maybe you can make a vid on that.
Just look at 0:55 he said it
It is the quadratic formula but with a=1 and b and c get renamed, the 2 that’s left in the denominator gets distributed into the root
What else should it be?
So you can divide by a and then use pq?
absolutely
yes
Yes
That is the whole idea behind the pq-formula.
Take p = b/a and q = c/a, plug them in, do the math and you get the abc-formula back.
It was called completing the square when I learned this in Detroit Public Schools. It works best when when the quadratic equation has a leading coefficient of 1.
So, it is taught in school. It's just not always called the "pq formula."
Side note: this method is just one of many ways to solve quadratics. No one should have been taught only one method.
Learned both.
We use the pq formula in Germany, but personally I prefer the abc formula since in some cases you have to devide by the factor next to x^2
Which generally is easier than memorizing the midnight formula. That's why the pq formula is taught.
did my Abitur in germany. Never heard about pq. So please don't generalise from your school (or Bundesland) to every school.
@@Engy_Wuck If you went to school in Germany, you should definitely have heard of the pq formula. It is the standard formula to solve quadratic equations.
@@Nikiokomaybe in your school or in your Bundesland.
Please remember that in germany schools are regulated by the state and therefore are not equal everywhere in germany, especially not in such details (the broader picture is regulated by treaties between the states).
So your "in germany" definitely is wrong. "In my part of germany" or "in my school" would be correct.
@@Engy_Wuck You learn the pq formula in each school and every state, since it is the standard curriculum which is also used in the school books. Maybe you learned the abc formula as well on top, but the pq formula you learned for sure. And the reason for that is quite simple: it is easier to memorize. So, more students will be able to solve quadratic equations. The drawback of having to transform the equation to the normal form first is considered less severe than having to memorize the more complicated formula. When I was in school, the teacher presented both, but suggested to memorize the pq formula. But in the end, it doesn't matter which one you use.
Wow! I've been solving the quadratic equation using a,b,c for over 60 years and have never been so confused by p and q. Just set a=1 and be done with it.
It is an identical formula. It just requires always dividing by a. The proof is very easy.
ax^2 + bx + c = 0
x^2 + b/a x + c/a = 0
let p = b/a and let q = c/a
x^2 + px + q = 0
x = (-p + or - sqrt(p^2 - 4(1)(q))) / 2(1)
x = -p/2 + or - sqrt(p^2 - 4 q) / 2
x = -p/2 + or - sqrt(p^2 - 4q ) / sqrt (2^2)
x = -p/2 + or - sqrt((p^2 - 4q) / 2^2)
x = -p/2 + or - sqrt (p^2/2^2 - 4q / 4)
x = -p/2 + or - sqrt ((p/2)^2 - q)
It is the quadratic formula with a division of a and redefining the division of b/a as p and the division of c/a as q. We also distribute the 2 divisor among the components with a quick squaring of the 2 to bring it inside the square root with another distribution inside the square root. It is a choice of doing the division first in all cases.
@@ronaldking1054 I was just joking. My brain has been programmed to ignore "a" when it is 1, but "a" is still in my mental image of the quadratic formula.
@@ronaldking1054The derivation of the quadratic formula (with a, b, c) is a lot easier using Sridhara's method. The advantage of Sridhara's approach over the conventional derivation (which starts by converting the quadratic into a monic quadratic equation by dividing both sides by a) is that this derivation avoids the use of fractions until the very last step. This is done by by multiplying both sides by 4a before completing the square at the left hand side:
ax² + bx + c = 0
ax² + bx = −c
4a²x² + 4abx = −4ac
4a²x² + 4abx + b² = b² − 4ac
(2ax + b)² = b² − 4ac
2ax + b = √(b² − 4ac) ⋁ 2ax + b = −√(b² − 4ac)
x = (−b + √(b² − 4ac))/2a ⋁ x = (−b − √(b² − 4ac))/2a
why did you private the new pq formula video :( I saw in like 3 times but I was listening to music and wasn't in the mood and when I finally clicked it (I left the yt open on my laptop) it was private :(
Oh no worries. I will upload it to here. I think it fits better here than the main channel.
Fun fact: the PQ formula just simplifies to the quadratic formula but isntead of 4ac its 4q and instead of 2a its juat 2
rare US math W: the pq-formula is an unnecessary variation on the quadratic formula that is slightly weaker because it increases the probability that a fraction will appear, and requires additional reasoning to be applied to the case where a ≠ 1, creating a failure point. if you're already doing algebra like this, 4(1)(-432) should not intimidate you.
With a trained eye, you can see that the p-q formula is "only" a simplification of the general quadratic formula with a = 1.
It is taught in school! In Romania I have learned it 35 years ago. Why would this gentleman imply that he found the secret of the plum's stem?
We don’t usually teach or use this in the US.
I love this expression, "found the secret of the plum's stem". I've never heard it before, but its meaning is very clear!!
I will be 100% real with you guys. The pq-formula is Germany's "classic quadratic formula". Never used the "normal" one in my life.
The abc formula is called the midnight formula. Because if your teacher, hypothetically, rang you up in the middle of the night, you should be able to recite it. But yes, in school the pq formula is taught. The drawback is that you have to divide by the coefficient of x² when it's not 1, but the big advantage is that the formula is easier to memorize, which means that more students will be able to solve the question.
@@Nikioko facts. Heard this from my college peers too. In Bayern bringt man den Leuten das aufjedenfall so bei, sonst ka.
Isnt it the same thing? Just setting a = 1. I don't see any noticeable benifit of using this...
It's exactly the same, the only real difference is that in deriving the classic quadratic formula you'd multiply out (p/2)² to get p²/4 put q to the same base giving (p²-4q)/4
And then when you square root the /4 part can be moved outside as /2 since the square root of ¼ is ½.
Really this is just another way of writing the formula.
My grandkids call it the new math, which is way over my head. Think about it. It's all about efficiency. Working with two numbers as opposed to three. (3/2)X100% equals 150%, so my grandkids get the answer 50% faster than I do, which is why they like competing against me.
@@roger7341 that is the case in for a monic quadratic but for the coefficient of x² ≠ 1, the pq formula works with fractions which is equally as ineffective as using three numbers.
@@arnoygayen1984 That's why he said at the end that one should only use the pq formula in certain simple cases, not always.
@@bjornfeuerbacher5514 in the simplest of cases, you dont need a formula at all. otherwise, if a=1 then you can simply omit it from the equation. pq is an unnecessary rearrangement for those who learned abc first
I'm speechless.
Mode 5 3 for life
Even with odd-numbers for b ... you can always use the pq if you are familiar with fractions! 😉
let b = 7 --> -p/2 = - 3.5 or better -7/2
then (p/2)^2 = p^2/2^2 = p^2/4 = 49/4 😉
In my country (Germany) we have never learnt the abc but pq only because it's far easier to memorize! And to all the people who think those formulas are different: Take the abc and let a = 1. Then abc becomes pq automatically! 😉 It calculates the other way around but still it is the same.
It had been weird when I saw the abc for the first time at school in Australia... and I could not understand why this difficult to remember formula was applied at all at school.
Never learned it that way. Seems a bit easier
Excelente
You cannot say x equals both 18 and -24. A much better notation is to give the answer x_1=18 and x_2=-24 which is logically correct. Otherwise you must use or (exclusive) for x=18 OR x=-24 and *not* both. If you answer like you did on an exam you will get subtracted points
The pq formula is much easier to memorize than the abc formula. The only drawback is that if you have a multiple of x², you have to divide the whole equation by that number first. That is an extra step, but an easy one in comparison to the abc formula. If you have a = 1, using the pq formula is a no-brainer, anyway.
U just need to put a=1 on QF, then devided by 2 numerator and denomonator, put ½ into square root, and tadaaa u have pqF.
If q is 0, the parabola described by y=x^2+px will go through the origin. Interesting, if you vary p, the minimum of this parabola will trace another parabola: y= - x^2
here in sweden this is the one we learn, not the quadratic
In Sweden they only teaach the PQ formula and not the quadratic formula, so whenever we get a quadratic equation you have to make x^2 stay alone
Just set a=1 and you can get this real easy
"But I'm not an American"
Just set p=b/a and q=c/a and you get back easy too
PQ and Quadratic are the same really. P is just b/a, and Q is c/a, and otherwise they are identical.
I learned the Quadratic formula, but honestly, I think the PQ is easier to remember because it uses fewer letters, and is the simple sum of the center point and the distance out to the intersection, rather than a big fraction with a sum in the numerator.
When p is even, better to use completing the square instead of memorizing new formula
just asking, if students were to use this equation even tho it wasn't teached, can they lose mark?
If they have a good teacher, then no.
But if the test states … using the abc-formula… than yes.
If you just have to solve a quadratic equation, you can use any valid method: midnight formula, pq formula, Vieta's Law, quadratic completion.
If you have to use a certain method, you lose points for not using this method.
Does it work when the coefficient of x is not 1?
If you have for example 3x^2+6x+9=0 you first divide the whole thing with 3 to get x^2+2x+3=0. Then proceed like in the video.
You divide the whole equation by a and there you go a monic quadratic
You can't use the formula but you can complete the square like that :
7x² + 35x + 3 = 0
7x² + 2×√7x×√7×5 + (√7×5)² = -3 + √7×5
(√7x + √7×5)² = -3 + √7×5
@david-ue6ed you put in the frattion into p and q
Then you need the rpq formula, which is really the abc formula in disguise. Just don't set a or r to zero or you no longer have a quadratic equation.
its cool but i like completing the square
he actually did read my comment on the other video wowwww
Is just like the quadratic by setting a=1
if you take the a away from the quadratic formula they are essentially the same, the /2 just took a light jog
I'd argue that the PQ formula is unnecessary. If you plug a=1 into the quadratic formula you'll get the pq formula.
Both the pq formula and the quadratic formula are proved in the exact same way. It's just that the QF is more generalized.
The pq formula is the one which is taught in schools around the world. Why? Because it is easier to memorize than the abc formula. And, as we see, we get smaller numbers during the calculation.
@@Nikioko you only get smaller number if a = 1 to begin with, or if b and c are integer multiples of a. In all other cases you now have fractions inside a root. Sometimes ugly ones.
Quick, solve 19x²+7x-12=0 with pq... Hint: you now have to square 7/38.
PQ formula. I had never seen this. In reality, except for the challenge, I would use an online quadratic formula calculator, just as I use a calculator or spreadsheet for most of my problems simply because it is fast and less error prone. Math is very nearly my "first language" and at age 73 I still enjoy learning something new.
is this just vertex form except you are using p and q instead of h and k. (I learned it vertex form is y = a(x-h)² + k. Where h is 1/2(-b/a) then k= -a (h)² + c. It kind of looks like that just different variables used to represent the vertex.
Instead of the derivation from 1:15 to 5:15, you could also start from the a-b-c formula, factor out 4a² under the root and simplify a bit to get - b/2a +- sqrt( (b/2a)² - c/a ). Then by setting p = b/a and q = c/a, you immediately have the p-q formula, and you have shown that both are equivalent.
Compute the Integral x^2024/(x^2+1)^2023 dx
K so all good and all but my math theacher brain will explode if i use this formula instead of quadratic formula😅, best math guy ever!!!!
I hear you. My grade school teacher barely graduated high school, so the abc's is all she ever taught us. One misses a lot when growing up in a one-room school house. And to think, sputnik was launched into space only a few years later. I'm sure they had advanced well beyond our abc's to the pqr's. Some of us old folk are way behind the times.
You have a _really_ bad math teacher if he/she does not grasp that. That's only basic algebra!
@@bjornfeuerbacher5514 so you might br surprised but our teachrr uses the guide book or basically thr answer key and dictate our answers to us, maths answers........
Always use the pq-formula. Even if a≠1, just make it equal 1 😂
It makes things much easier.
(Yeah, I'm also from Germany and the quadratic formula was shown to us, but we never used it)
2 formulas? That's too complicated - Einstein :-)
See it as another arrow in your quiver. Another tool in your tool kit. You can then decide which is the best to use in your individual case.
i love you bprp 🥵
3 days ago?😟
Bro what
same
what the hell
1:13 but whyyy sirr??
Why so much calculation you can put a=1 in quadratic formula to get pq formula
true but in Deutschland we just teilen durch a, okay?
the p-q formula is litterally just the quadratic formula where a = 1
Mind your Ps and Qs.
Just one honest comment before I click the necessary buttons to ensure you I'll not bother you again: There is a level of mathematics that most people will never need and will never use, but people study this math anyway because it is required in school.
I honestly will never need a quadratic equation for anything in real life, and .99999....does functionally equal 1 in every case except when a difference of .00000... 1 matters.
Thank you, and best of fortunes.
At 0:28 he says if a=1 then the formula will be more intimidating. I don't understand
It’s exactly the same formula. We call it reduced form.
you MIGHT have convinced me by using an equation which is not factorable.
Same number of components in the formula, but if the x^2 coefficient is not 1, you need to divide through and then execute this formula with fractions. Why make it harder?
5x^2 + 7x + 12 = 0. Youare busy squaring and manipulating 7/5 and 12/5. And squaring 7/10. Why?
So P is b, q is c and a=1....
Ye we in germany have like luck or something idk
8:40 If p is an odd number, you have to square a number which ends in .5. That's quite easy to do: (n + 0.5)² simply gives n(n+1) + 0.25. No need to use fractions.
Distinction without a difference. It's just a simple change of variables that is easily derived from the traditional quadratic formula by inspection. There are other approaches that truly start differently.
Good morning, Mr. "presenter" of this block!
You want to add something unnecessary, because the reasons are that it does not help the student at all, since he has to do unnecessary calculations if p and q are fractions, it seems to me that certain things have been mixed up. In reality it would be necessary to make the calculations simpler, if b and c are divisible by a.
It has no methodical value if the operations to be carried out according to any path become complicated.
I hope my comments don't bother you, if you really take them into consideration.
If you want to collaborate with something when trying to do mathematics, you must keep in mind that it really has a didactic and methodical value in which the student becomes aware that he or she has learned something new about the topic, in this case, in the solution. of a second degree equation.
Good day!
This is not unnecessary, that is the quadratic formula how it is taught all around the world. And why is it taught, rather than the midnight formula? Because it is way easier to memorize for students and will deliver the same results.
And if you know both formulas, see it as another arrow in your quiver. You can then choose the formula which is more suitable for the problem.
Pq-formula is not better than the traditional one. I'd rather better the classic one taught here in Brazil. And a better way to demonstrate it... Is substituting both p=b/a and q=c into the classical solution.
It works better when p is even. 😃
Well there's is nothing new with this, just a = 1,and both the formulae are same
...why? i dont see the point, quadratic formula just has more use
He explained it in the end: If the coefficient of x² is 1 and p is an even number, the p-q formula is _very_ easy to use, much simpler than the a-b-c formula. And it's easier to memorize.
The pq formula is much easier to memorize and uses smaller numbers. The only drawback is that you have to divide the whole equation by a if a is not 1.
You have a single gray hair near squamous suture 😪
I don't really see the difference... they're the same formula, you just don't see A, and the 1/2 is distributed.. you can do that (distribute the 1/2) while using the regular formula, as well, without memorizing another version
He explained it in the end: If the coefficient of x² is 1 and p is an even number, the p-q formula is very easy to use, much simpler than the a-b-c formula. And it's easier to memorize.
@@bjornfeuerbacher5514And even if a isn't 1, you can divide the whole equation by 1 and use the pq formula. Which is still easier than memorizing the midnight formula. And in times of calculators, you should be able to calculate p²/4, even if p is an odd number.
@@NikiokoYou probably mean "divide the whole equation by a"? Err, thanks, that's obvious, you don't need to tell me that. I even wrote that myself several hours ago in another comment.
And this is _not_ about using calculators, but about doing calculations by hand.