Viral question - solve this for the ATM card's PIN code

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A joke note offers use of an ATM card, but only if you can can solve an impossibly hard integral.
0:00 problem
0:53 solution
5:33 numerical
Math StackExchange Claude Leibovici
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Euler substitution
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Reimann sum
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Midpoint rule
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09glasgow09, CC BY-SA 3.0 , via Wikimedia Commons
solution @NathMathTutoring
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27 июн 2024

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Комментарии : 321

@laszloliptak611 6 месяцев назад

Nice solution. I was a bit disappointed that the final result is not a 4-digit integer.

@BenDRobinson 6 месяцев назад

I felt the same! This puzzle ticks that box - sum(n=0 to infinity) of (n^6)/(2^n) but it's not nearly as exciting

@mabl4367 6 месяцев назад

I agree. I was expecting a positive four digit integer.

@fadz5210 5 месяцев назад

... and ans. is negative@@BenDRobinson

@aldoushuxleysghost 5 месяцев назад

Just stipulate that you must round your answer to four significant figures to get your code

@w8stral Месяц назад

How could it POSSIBLY be a 4 digit integer when x is the LIMIT of 0-->1??? Ai carumba dude

@smylesg 7 месяцев назад

9:27 "But these are questions above my pay grade," said everyone watching this channel at least once.

@pi-meson7677 7 месяцев назад

I have a simple(but calculative) way: The numerator 3x³-x²+2x-4=(x²-3x+2)(3x+8)+20(x-1) Also x²-3x+2=(x-1)(x-2) Thus original integral breaks into 2 standard integrals of the form (3x+8)sqrt(x²-3x+2) and 20sqrt((x-1)/(x-2)) Both of these can be solved by substitutions.

@chrisjohnson8666 6 месяцев назад

That's what I did and got your result. I stopped there as it was clear there was still a lot of work ahead. Good to know that I got that far without an error. I thank you for that.

@Slash4608 5 месяцев назад

@@chrisjohnson8666For the first part,put numberator = A(derivative of denominator) + B. Solve for a and b and put that in place of numberator. First part is solved and second part is basic level

@chrisjohnson8666 5 месяцев назад

@@Slash4608 True dedication. If I need to solve this equation I will return to your advice. 🙂

@teambellavsteamalice 2 месяца назад

The important thing about this comment is that you should always look to simplify. If the maker of the problem had put any effort into it, chances are things will cancel out.

@docsigma 7 месяцев назад

“Thank you darling. Unfortunately I entered the PIN code incorrectly too many times, so the machine ate the card”

@j.u.4.n620 7 месяцев назад

Imagine getting pincode as -pi/e😂

@huzefa6421 7 месяцев назад

😅

@sunrevolver 7 месяцев назад

If we ignore the - and /.. I think it'll be 744433 😅😅😅

@kavinesh_the_legend 7 месяцев назад

In my calculator I got the answer as -ve 1.1557

@Sugarman96 7 месяцев назад

That's just -1

@justaguywhoisanidiot159 7 месяцев назад

@@Sugarman96Engineer ?

@karhukivi 7 месяцев назад

When you crack this one, you are so mentally fatigued that you just order a takeaway and eat it alone...

@theupson 7 месяцев назад

since x-1 is a factor of the top, you can rewrite the integrand as (3x**2 +2x+4)( (x-1)/(x-2) )**.5 riffing on the theme of the euler subs, you set the square root equal to 1+t. because of the reduction of order, the algebra is materially less painful.

@blessedslave 5 месяцев назад

Both you and the Wikipedia page at 1:24 made the same typo. Interesting

@kylestephenson7024 5 месяцев назад

@@blessedslave he didn't make a typo though?

@1st2nd2 7 месяцев назад

I consider "working things out by hand" to be the mathematician's equivalent to "getting your hands dirty" when building things. Yes, one can use tools (Wolfram Alpha) to work things out, but there is something fundamentally satisfying about having mud to up your elbows and getting in there to figure out exactly _why_ something is that way.

@inplfw 7 месяцев назад

Even though I understand the problem, it's been ages since I've used the actual math to solve it. Typing "integral_0^1 (3x^3-x^2+2x-4)/sqrt(1x^2-3x+2)dx" into WA Is understanding how to use a tool. I wouldn't build a house with a hammer if it left the framing nailer gathering dust. Two types of people, I guess.

@Zarunias 7 месяцев назад

You don't even need Wolfram Alpha. My calculator can do it. (Although I have to admit it's a fancy one (TI-92)).

@1st2nd2 7 месяцев назад

@@Zarunias i remember using that when it first came out. A nice little machine :)

@1st2nd2 7 месяцев назад

@@inplfw i understand that perspective.

@TheAgamemnon911 7 месяцев назад

It depends on whether the problem is just some busywork thrown your way or actually interesting - at least for you.

@lofinightc0re36 7 месяцев назад

trig substitution is actually easier using the reduction formula for sec^nx. factorizing the numerator and the denominator and putting sqrt(x-1) = u the denominator simplifies to Sqrt(u^2-1) and then substituting u = sect , du = sect*tant dt, the tant forming in the denominator will cancel out the one with the dt and the integral with just be some sec^nx (using a calculator it is exactly this 3sec^7(t) + 8sec^5(t) + 9sec^3(t) dt and this is so easy with the reduction formula as it is just odd powers.

@bjornfeuerbacher5514 7 месяцев назад

I did it in a similar way, but used u = cosh(t) instead.

@smilerbob 7 месяцев назад

Thank you for your card hunny, I didn’t need the PIN number in the end as it has contactless. We all enjoyed lunch and I have updated all the clothes in the wardrobe All my love Your Darling xx

@ernestregia 7 месяцев назад

Imagine finding out that the card balance's zero after you managed to crack down the password 😂

@user-gb6sz3ve4f 7 месяцев назад

-2.981 is also its initial balance

@amoswittenbergsmusings 7 месяцев назад

I used Maxima. I defined the integrand as f(x), then used integrate(f(x),x) to get the antiderivative, used the at() function at the limits and got of course a complex answer because of the logs of negative values, but expand(float(%)) gave the right numerical answer. For the sake of completeness here is the antiderivative according to Maxima: (135*log(2*x+2*sqrt(x-2)*sqrt(x-1)-3)+sqrt(x-2)*sqrt(x-1)*(16*x^2+52*x+202))/16

@nripdave673 7 месяцев назад

he is intelligent and logical lover......😆😆😆😆

@buxeessingh2571 7 месяцев назад

Huh? Even though it was not in the syllabus, I always taught Euler substitutions when I taught Calc II. I also taught the equivalent alternative of completing the square, using a trig substitution, using the z = tan(t/2), and then using a partial fraction decomposition.

@theupson 7 месяцев назад

it's not clear to me that the obscure approach works any better than the obvious one. the bounds are a hassle in trig sub, but the elided algebra in the vid is pretty heinous. have you got an example handy where euler sub is actually an improvement? i do note the 3rd euler sub is more user friendly for this problem; enough so in fact that i'd be tempted to try it in preference to the other two even with complex roots.

@lofinightc0re36 7 месяцев назад

@@theupson trig substitution is actually easier using the reduction formula for sec^nx. factorizing the numerator and the denominator and putting sqrt(x-1) = u the denominator simplifies to Sqrt(u^2-1) and then substituting u = sect , du = sect*tant dt, the tant forming in the denominator will cancel out the one with the dt and the integral with just be some sec^nx (using a calculator it is exactly this 3sec^7(t) + 8sec^5(t) + 9sec^3(t) dt and this is so easy with the reduction formula as it is just odd powers.

@shroomer3867 7 месяцев назад

The card owner thinking he is smart: 🤓 WolframAlpha: 🗿

@ObsidianParis 7 месяцев назад

"Honey, I couldn't figure out your PIN number so I ordered everything online instead. Luv you ♥"

@daveh4334 7 месяцев назад

Dammit, I need a new PIN now...

@BenDRobinson 7 месяцев назад

I wonder if the composer of the original joke knew or cared whether the integral could be solved. Pretty irrelevant to the joke after all. I think it's just a little bit unsatisfying that the answer wasn't actually a number in the range 1000-9999.

@BenDRobinson 6 месяцев назад

Taking my lead from another video on this channel, here is a puzzle the joker could have used that does have a positive 4-digit integer solution: sum(n=0 to infinity) of (n^6)/(2^n) It's not nearly as gnarly as the integral of course, and even easier to do a numeric hack in Excel... and it's clear what it's converging to after only 40 terms. I think it's still a cool challenge for someone to come up with a good tricky integral like in the original, but somehow ensure a suitable answer. And for bonus points, use a function that behaves very badly near some asymptotes, verging on pathological, to make a computational approach as hard as possible.

@ejn1011 7 месяцев назад

3:23 "Now comes the hard part" 😲 I thought we already had the hard part!

@muralinagarajan8305 7 месяцев назад

Yes, you read my mind ..😊

@sanjoychandrakarmakar712 7 месяцев назад

Oh no the wife saw your video

@Nxck2440 7 месяцев назад

I managed to do it using more common methods, complete the square in the root and use the trig sub x - 3/2 = 1/2 sec theta. It takes a while but just to show you can do it without knowing this special Euler substitution. You need to know integral of sec^3 theta using this though.

@bjornfeuerbacher5514 7 месяцев назад

I used a substitution with cosh instead, I think that leads to even easier integrals.

@yashmehta9299 7 месяцев назад

Can actually be done with simple algebraic and trigonometric substitutions, and integration by parts.

@divyeshkatariya4951 7 месяцев назад

Imagine she finding this video right away and skiping until end to see the answer.

@xwyl 6 месяцев назад

In all conceivable way we can know that he doesn't love you, and he never dreamed you could get the pin.

@lover123777 7 месяцев назад

Intelligent and logical ways thanks

@andreasr8033 7 месяцев назад

I think your integral is not well defined as the denominator gets zero for x=1 and this is part of the intervall you integrate. At least it is a remediable gap in definition as the counter gets zero at x=1, too. Maybe a limit notation to be more exact?

@Whynot690 7 месяцев назад

Me: "I'll never use calculus in my everyday life" My teacher:

@joeschmo622 7 месяцев назад

I used Simpson's Rule, but unfortunately it was Homer Simpson's Rule and I got the wrong answer.

@carultch 6 месяцев назад

An idea for a Mathologer video: explain Simpson's rule in a way that Homer Simpson could understand.

@gregheffley6400 6 месяцев назад

split the integral as follows ((3x^3-x^2)-(1)+(2x-3)) the denominator remains same. Now the main challenge is (3x^3-x^2) so split it again as follows 3/2[x^2(2x-3)+7/3(x^2-3x+2)+7/3(2x-3)] again main issue is to solve x^2(2x-3) the other 2 are just standard results after substitution, now for x^2(2x-3) use by- parts to get 2x^2*sqrt(x^2-3x+2) + int(x*sqrt(x^2-3x+2))----> rewrite this as in(sqrt(x^2-3x+2))+2x-3)/sqrt(x^2-3x+2) now this is another standard substitution problem (make sure to keep track of all the coefficients as one little mistake can change the whole answer) anyways this was my attempt of solving it, not the shortest but still got the answer right, Cheers!!

@steveschooler5840 6 месяцев назад

Alternative : int {sqrt[ (cx + d)^2 - a^2]}^{-1} dx conquered via cx + d = a sec t implies c dx = a sec t tan t dt. Using [sec t]^k = [2x - 3]^k, you then convert the numerator into a 3rd degree polynomial in [sec t]. Then, you will need formulas for int sec^k t dt : k in {1,2,3,4}.

@43.rohitsharma76 7 месяцев назад

i think we can do it like this as well it follow first split denomiator with x2 and x3 terms and other with 2x-4 term then we can solve this linear by quadratic term easily now with 3x2-x2 take common x2 then apply same linear by quadratic with linear factored term and then write it like x2(a d/dx(denomiator +b) then multiply and again split then first term can be solved easilty with some substiution and other term by adjusting -3x+2 and 3x-2 first term of this will be (ax2+bx+c)^1/2 kind of stuff and other one linear/quadratic again and its done !!! i hope it helps ;-;

@tanmayarora8730 6 месяцев назад

Mom's always use their birthday as pin 😂

@StructuredChaos 7 месяцев назад

you have just explained Finite Element Analysis in 10 minutes.. :)

@Fakepokeemon 7 месяцев назад

She finally cracked the pin, But unfortunately there's no money in his bank account 😅

@JekyllandHyde_ 7 месяцев назад

3:08 Tedious. I see what you did there.😏

@user-uh9bo2im1h 5 месяцев назад

These ones are always easy to solve as long as there is a quadratic in the bottom you have to complete the square first then you make ur u-sub so that the structure on the bottom is sqrt(u^2+c) then u use u = sqrt(c)*sin x then you’ve gotten rid of the denominator and are left with a whole bunch of trig functions for this quest to the max power of.3 which is easily solvable

@JohnDlugosz 7 месяцев назад

re hacking in Excel: My pocket calculator in the '90's included numeric integration. I would have thought that Excel would include a function that does all this automatically, using something more advanced like Simpson's rule (IIRC).

@rogerkearns8094 7 месяцев назад

He shouldn't have left the formula written down, anyone who saw it could have stolen the money. ;)

@holyshit922 7 месяцев назад

sqrt(x^2-3x+2)=u-x substitution should give the result After this substitution you shoul get power function to integrate

@rameshhariharan2623 6 месяцев назад

Great, i learnt lot sir.

@mathmannix 7 месяцев назад

I really thought it would be something that evaluated to pi. Because that would be easy to remember, as well as potentially hard to figure out.

@vidlover7875 5 месяцев назад

That moment at 3:23 when my brain is trying to keep track of this extremely convoluted train of thought and he goes "So now comes the hard part" and I was just about to scream "You mean this isn't even the difficult part?!"

@darske1 7 месяцев назад

The wife proceeds to use Wolfram Alpha

@thatdozy 7 месяцев назад

0:53 "So this is not something that's ever taught in a normal classroom..." I expected him to say Photomath

@eighteenin78 6 месяцев назад

I would take a screen shot of the curve, colour in the area bounded by the curve and the x and y axis a bright pink, then write a program which counts the number of bright pink pixels in the image.

@okaro6595 7 месяцев назад

This can also be done with many calculators but as it is an improper integral one needs to put 0.99999 or something like that as the upper limit. There are also calculator apps for a phone-. While my calculator calculates it for 15 second the phone gives the answer immediately when one enters 1 as the upper bound (no need to press =)

@aarondriscoll287 7 месяцев назад

The TI-83/84 series of graphing calculators implement a Gauss-Kronrod quadrature that doesn't actually evaluate the function at the boundary points, so there's no need to set the upper limit to 0.9999 instead of 1.

@mh017509 7 месяцев назад

Now think that the Calculus (or, more correctly the Mathematical Analysis) was developed in the 18th century, including Euler's substitution.

@Spudz76 4 месяца назад

This is when you simply go into Walmart and separately purchase four sticks of gum with $100 cashback each time and walk away with $400 cash and don't need the PIN whatsoever. Although the newer systems don't seem to want to allow debit purchase for under $50 totals. Otherwise you cancel PIN entry and select "authorize" instead. Or buy $400 worth of junk , then immediately return it at the service counter, they will give back cash for any debit purchase. If you have a non-debit ATM card what decade are you from?

@ShyamSharwan 7 месяцев назад

Just input that integral in wolfram alpha🤣, it barely took 1 second

@JulianShagworthy 6 месяцев назад

When I put this into my scientific calculator precisely as it's formatted in this video, it spat out the answer '-19.328...'

@BassemFanari 6 месяцев назад

No need for Euler's Substitution here, it's too complicated. Just complete the square in the denominator to: ½ √((2x-3)²-1) Then use a Trig Substitution: Let sec(θ)= 2x-3 sec(θ)tan(θ)dθ= 2dx x= ½(sec(θ)+3) x²= ¼(sec(θ)+3)² x³= ⅛(sec(θ)+3)³

@mrespanfanx 6 месяцев назад

Getting that note is the moment you know you need to move on 💔

@nychold 7 месяцев назад

It's funny...the first thing I did was factor the top and bottom, to see if basic u substitution would work. It didn't, but I learned the bottom has roots of 1 and 2, which made me question whether there even was an answer.

@teambellavsteamalice 7 месяцев назад

The top equation can also be factored into x-1, but not sure it helps. This leaves: 3x^2 +2x +4 With x=t+1 you get g(t) = 3(t^2+2t+1) +2(t+1) +4 = 3t^2 + 8t +7 combined you get: t*g(t) / sqrt( t(t-1) ) Unfortunately this doesn't make it any easier, the creaters of the problem didn't choose g(t) as anything nice.

@theupson 7 месяцев назад

the soundest answer is just be fastidious with your improper integrals, and it will come out in the wash. but the 30000 foot view is that if the integrand is "like" (x-1)**-.5, the antiderivative will be "like" (x-1)**.5 ie harmless.

@lofinightc0re36 7 месяцев назад

trig substitution can helpful after that. factorizing the numerator and the denominator and putting sqrt(x-1) = u the denominator simplifies to Sqrt(u^2-1) and then substituting u = sect , du = sect*tant dt, the tant forming in the denominator will cancel out the one with the dt and the integral with just be some sec^nx (using a calculator it is exactly this 3sec^7(t) + 8sec^5(t) + 9sec^3(t) dt and this is so easy with the reduction formula as it is just odd powers.

@muralinagarajan8305 7 месяцев назад

@nychold, same here...😊

@THall-vi8cp 7 месяцев назад

"1, 2, 3, 4, 5? That's amazing! I've got the same combination on my luggage!!"

@mdasadrizwan9744 7 месяцев назад

Imagine inserting some power to differentiation 💀

@adrien5568 7 месяцев назад

Enjoy this bit of maths: en.wikipedia.org/wiki/Fractional_calculus

@Jack_Callcott_AU 7 месяцев назад

I wouldn't like to boast, but I will. I put this integral into my incredible HP50g calculator, and I got the exact answer , not a numerical approximation. I was using the exact mode, so I kissed the calculator.

@somnathkundu7354 5 месяцев назад

Could be even done Little bit more elegant. The denominator factors into sqrt{(x-1)(x-2)} and the numerator also give those terms. So it resolved to more manageable form.

@w8stral Месяц назад

Literally engineers NEVER use calculus. We always use numerical analysis and have done so since the advent of computers.

@kevinvanhorn2193 5 месяцев назад

Mathematica gets the symbolic answer in an instant.

@weylguy 5 месяцев назад

How many random Monte Carlo points would you have to throw at the plot shown at 5:39 to get the answer?

@FranFerioli 7 месяцев назад

The Euler substitution is a trick. You use it only because you already know it works in this specific case. Riemann integration is a much more fundamental piece of knowledge and numerical methods a much more versatile tool.

@roger7341 7 месяцев назад

Ha! The last laugh is on him because she's been programming the trapezoidal method.

@maxtonuponry 6 месяцев назад

"So, leaving the joke aside, how do you calculate this integral" is such a way to start a video. And yet, here I am.

@celkat 7 месяцев назад

"P.S.: Desmos is your friend honey"

@johnlee5709 5 месяцев назад

Hi Presh, l enjoyed the Excel solution. Thanks for the post.

@christoferchavanne7166 5 месяцев назад

Perhaps the whole point of including the value for x=1 (to make for invalid numbers, division by zero) was to show that the code itself was invalid. 😅

@wolfVFV 7 месяцев назад

3:20 so now comes the hard part THAT WASNT THE HARD PART?

@prateek746 7 месяцев назад

Imagine if ramanujan see this question 😂

@nicholasscott3287 6 месяцев назад

Pretty sure the reason why the last row broke is because it was trying to do calculations with an empty cell. You should have continued the first column another row.

@thorntontarr2894 7 месяцев назад

Another opportunity to show your viewers that numerical methods are powerful because real systems typically MUST be numerically solved, i.e. Climate Models, Weather Predictions and Fluid Dynamics.

@jackaw1197 6 месяцев назад

I saw this and immediately thought I could numerically solve it in minutes using matlab (or substitute the scientific programming language of your choice), especially since the result doesn't need to be exact, but just accurate to 4 or 5 sig figs

@tamararoberson8060 6 месяцев назад

@@jackaw1197 WolframAlpha is handy. I used it a couple days ago to find "2^63 nanoseconds in years" to justify not dealing with the rollover of a 64 bit integer (we just stop incrementing). WolframAlpha let me know that it's equal to "1.5 x approximate time one would have to lie supine before a bird would poop in one's mouth (≈200 years)". Windows calculator or Matlab would not have told me that.

@TheInfidel_SlavaUA 6 месяцев назад

needless to say this man is now maidenless

@edsimnett 7 месяцев назад

Fun! After many more years of excel "coding" than real maths, I tend to do this sort of thing a lot, would not have thought of it here. Question- what did you need column C for in the excel, it's duplicated by column E, which is what you use?)? [edited]

@AmazinCris 7 месяцев назад

column D calculates the midpoint (or average) between two x values so that it can be substituted into the column E (which is then used in the next column to multiply by the difference between the two x values). hope this answers your question!

@edsimnett 7 месяцев назад

@@AmazinCris My bad, I meant column C (column C and column E are the same formula)

@AmazinCris 7 месяцев назад

in that case, he might have just written column C to get the function by itself so that he could substitute in the value of the midpoints. @@edsimnett

@yurenchu 7 месяцев назад

@@edsimnett They are the same formula, but applied to a different variable (namely, one uses the cell in column B, the other uses the cell in column D). However, yes, column B is unnecessary, as the values in column B aren't used for anything.

@orisphera 6 месяцев назад

What do you get if you try to do this in Calc?

@edwardnedharvey8019 7 месяцев назад

Or just plot the function on a graphing calculator, and get the area. Easy! 😅

@nightmayor6114 6 месяцев назад

Imagine if the answer comes in form of tetration

@scottwalker8021 4 месяца назад

While the Euler Method will work, hyperbolic and secant-tangent substitution are a bit less of a slog.

@anketmohadikar8767 7 месяцев назад

*Plugs the integral in Wolfram alpha.* Bro must be punching air after that.

@Loots1 7 месяцев назад

How do i put in a negative pin code

@BronzeDragon133 6 месяцев назад

Easiest answer: Dial honey, "Give me the pin code without the tricks or you'll be sleeping on the couch by yourself for the foreseeable future." "2981." "Good boy."

@vectorexpanse 6 месяцев назад

Presses cancel on PIN pad to run as credit instead, bypassed!

@ghost307 6 месяцев назад

So is the PIN 2918 or 2981? The 2 methods return different results.

@nonamenoname6921 6 месяцев назад

Look again.

@LeBeaBae 7 месяцев назад

lmao i love this problem. I've done it 30 times

@davidseed2939 6 месяцев назад

simpsons rule may be better even with the same number of calculations. For the 101 data points you used the trapezoidal rule with weights (1,2,2,2….2,1 )/200 For Simpsons rule you would use weights (1,4,2,4,2,…2,4,1)/300 in both cases you are taking a weighted average over the range. this answer needs to be multiplied by the range(x2-x1) to give the required area (integral)

@maxpoweroverdrive 7 месяцев назад

I mean this is a cleaver way to say no to spending excessive money...

@carultch 6 месяцев назад

All she has to do is know how to use a graphing calculator, and the area under the curve feature. You don't even need to know the methods Presh showed to do this.

@bernhardriemann3821 5 месяцев назад

honey outsmarted the husbanded by using numerical methods

@Terminarch 6 месяцев назад

Just swipe the card. I haven't been asked for my pin in literally YEARS. Those chip things are weird.

@mandolinic 7 месяцев назад

The PIN code is quite clearly 1435

@Cryscorde 7 месяцев назад

“Solve the following puzzle to get the pin- what are you doing? I haven’t gave you the puzzle yet.” Me, brute forcing every combination: “Trust me, this is quicker.”

@-igor- 5 месяцев назад

Greetings from 2024! I like very much the Excel solution.

@bob456fk6 7 месяцев назад

I keep my PIN code on a piece of paper in my wallet. It's a fairly simple function requiring only a 4-function calculator to evaluate. 🙂 It doesn't look like a PIN code to the casual observer.

@forbidden-cyrillic-handle 6 месяцев назад

Ok Bob. I'm no longer a casual observer.

@mohamedhajali3829 7 месяцев назад

It's my first time to ignore the solution, I didn't need from ATM card . It's easy to brake the machine than solving the math problem 😂😂

@tobyfitzpatrick3914 7 месяцев назад

Our Ancient Mathematical Forefathers didn't have ATM cards, so it's a wash...

@Zzzbello 7 месяцев назад

me after solving the question slightly wrong and getting arrested for commiting fraud

@jige1225 7 месяцев назад

🤣🤣🤣👍

@PivDen-jv3th 7 месяцев назад

We learned this method on Calculus 2-3

@sapito169 7 месяцев назад

gentelmans never do it woman will find another man who can make the integral for her

@DeveloVooshGWeb 7 месяцев назад

encoding messages without ciphers be like:

@rodrigomarchi9755 7 месяцев назад

Great vid

@birandkoray 7 месяцев назад

why t is plus root 2 for x=0, minus root 2 is also a solution 2982 may be the answer, too when rounded considered

@Ninja20704 7 месяцев назад

You have to refer to the original definition of the substitution. sqrt(x^2-3x+2)=x+t Sub in x=0 you get t=sqrt(2) The -sqrt(2) is extraneous because we squared both sides to isolate x.

@birandkoray 7 месяцев назад

satisfied , thnx@@Ninja20704