Math Olympiad | A Nice Algebra Problem | How to solve for "a" and "b" in this Problem?

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Күн бұрын

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@giannaleoci2328 6 ай бұрын

a^2-b=73 b^2-a=73 a^2-b=b^2-a a^2-b^2=b-a (a-b)(a+b)=-(a-b) a-b div 0 a+b=-1 a=-b-1 b^2-a=73 b^2+b+1-73=0 b^2+b-72=0 b=[-1+-rq(1+288)]/2 b=[-1+-17]/2 b1=-9 b2=8 a=-b-1 a1=9-1=8 a2=-8-1=-9

@is7728 6 ай бұрын

Smarter method (gave it a like)

@garrettvanmeter5831 6 ай бұрын

Again, only one solution is needed since you can appeal to symmetry to get the other

@is7728 6 ай бұрын

@@garrettvanmeter5831 Agree

@RoK2016 2 ай бұрын

You and Admin only solved for one condition: a+b+1=0. I call your result is solve1 and 2. Other condition is a-b=0. a^2-a=73, a^2-2×1/2×a+(1/2)^2-rq(73x4+1)/2=0, so a-1/2-rq(73 +1/4)=0, or a-1/2+rq(73 +1/4)=0 sovle 3. a=b=9.058621 Sovle 4. a=b=-8.058621

@MayorKing013 2 ай бұрын

@@RoK2016the conditions in the problem statement say that a is distinct of b, so is no necessary to solve in that case.

@ЛидийКлещельский-ь3х 6 ай бұрын

Hellooooo 😊). A bad solution to an easy problem. After 2:40 we substitute (1) a=b and (2) a=-b-1 in (0) a^2-b-73=0 . We get : (0,1) b^2-b-73=0 , and (0,2) b^2+b-72=0 . Therefore : (3) b1=[ 1-sqrt(293) ]/2 , b2=[1+sqrt(293) ]/2 (4) {thanks to Vietta } : b3=-9 , b4=8 , we substitute (3) in (1) and (4) in (2) . We get the right answer. With respect , Lidiy

@ashtongreenidge 6 ай бұрын

Recall the condition that a is not equal to b.

@fauzilvaleev7904 Ай бұрын

I have no idea in maths, but I’m admire your manner of explanation! Thank you, Sir!

@learncommunolizer Ай бұрын

Wow, thanks You are welcome!

@fauzilvaleev7904 Ай бұрын

@ I’m Russian. Science, progress, development aren’t empty words for me. At my school 50 years ago haven’t been explained properly. I see your love for the science and maths, which explains us how is created and exists our small Earth . Thank you very much.

@tonyennis1787 5 күн бұрын

I agree, this guy's presentation is top tier. Doesn't waste time with obvious arithmetic and copying down the same line over and over with tiny changes.

@robertliu3176 3 ай бұрын

Since a + b + 1 = 0 then substitute b = -a - 1 into eqn 1 and get a^2 - b = a^2 + a -72 = 0 => (a-8)(a+9) = 0 => a = 8 or -9 , then b = -a - 1 = -9 or 8. It's symmetric.

@tonyennis1787 5 күн бұрын

At 4:50 you conjured a+b=-1, which was well done. However, you could have saved a lot of time by just going back to equation 1 and substituting b-=1-a for b. You end up with (a)(a+1)=72 which is easy enough to solve. Much less error prone than 7 more minutes of algebraic gymnastics.

@margotconard7644 5 ай бұрын

a^2 = 73 + b b^2 = 73 + a the nearest square of an interger that is larger than 73 is 81 by inspection if a equals 9 and b equals -8, the problem is solved.

@爷爷-m2p 5 ай бұрын

应该是a=-9,b=8或a=8,b=-9吧，你这种解法我们也有类似的，这种解法在中国被称为“瞪眼法”我们写成: 注意到a=-9,b=8是一组解，显然a=8,b=-9是另一组解，证毕。

@ika5666 5 ай бұрын

It took me around 10 sec to find the solution. Glory to Ukraine.

@丁爸爸-w3f 5 ай бұрын

@@爷爷-m2p 你认为谁能看懂汉字？

@oahuhawaii2141 3 ай бұрын

But you're assuming the solutions are integers. They can be other types of real numbers, or even complex. The other 2 solutions are a = b = (1 ± √293)/2 ; they work for the thumbnail, before he added the a ≠ b requirement to his video.

@oahuhawaii2141 3 ай бұрын

@爷爷-m2p: That's due to the symmetric system of equations. Variables a and b are interchangeable.

@ernestboston7707 2 ай бұрын

I simply tried numbers near the square root in my head, thinking that one needed to be negative, and tried -9 & 8, and it worked, so it was just about a minute of work.

@tassiedevil2200 Ай бұрын

from the symmetry of the equations, if (p,q) is a solution, so is (q,p). Also, for a and b different, once you find that a=-(b+1) then might expect a and be will be integers and it is pretty immediate to see that (8,-9) and (-9,8) are the solutions. I also found it a bit obscure to always just verify using equation (1).

@soniamariadasilveira7003 6 ай бұрын

I loved this problem!

@ina-j2p 4 ай бұрын

辺々引くとa^2-b^2=b-a a+b=-1 a^2+a+1=73 a^2+a-72=0 (a-8)(a+9)=0 a=8,-9 b=-9,8 ∴(a,b)=(8,-9),(-9,8)

@zenziryx3242 Ай бұрын

My thoughts after 15seconds of video: I can guess that a=8 and b=-9. Solving x^2-y=z where z is 73, the only x^2 that could be higher than 73 is 81. Thus 81-8=73. Then solving y^2-x=z using X as either +9 or -9, I realized that it can only be a=8 and b=-9 b/c (8^2)-(-9)=64-(-9)=64+9=73. If there were to be a similar yet more complex problem; I bet it would written as (a^2)+a=(b^2)+b=72=xy where a>b and x=(cbrt(a))^(sqrt(-b))=y+(cbrt(a))-(sqrt(-b)) while y=(sqrt(-b)^(cbrt(a)) … or something like that (>w

@johnplong3644 4 ай бұрын

You MUST well organize and neat when doing this problem

@mzallocc 5 ай бұрын

a^2-b=73 ... (1) b^2-a=73 ... (2) (1)-(2) -> a^2-b-b^2+a = 0 a^2-b^+a-b=0 (a-b)(a+b)+a-b = 0 (a-b)(a+b+1) = 0, since a not equal to b, a+b+1=0, a=-b-1 .. (3), Putting a=-b-1 into (1), (b^2+2b+1)-b=73 b^2+b-72=0, solving this -> b=-9, 8, and using (3) a=8, -9 respectively. (a,b)=(8,-9) and (-9,8)

@evgenysapotnitsky8234 5 ай бұрын

Задача, на самом деле, очень простая. Вычитаем второе уравнение из первого и после преобразований получаем (a-b)(a+b+1)=0. Отсюда следует a=-1-b (a не равно b по условию задачи). Подставляем полученное выражение для "a" во второе уравнение и получаем b^2+b-72=0. Решаем квадратное уравнение и получаем ответ: 1) b=8, a=-9; 2)b=-9, a=8. Тринадцать минут на такую задачу - очевидный перебор))

@evbdevy352 5 ай бұрын

Я тоже решил именно так,отнял от(1) (2).Получил такое же кв.урав-е.

@evbdevy352 5 ай бұрын

Потратил не более 5 мин.На Олимп.ге тянет

@НадеждаМаксимович-ф3е 5 ай бұрын

Это система со звездочкой. Для олимпиады простовато. Для детей, котопые любят шевелить мозгами очень интересное.

@pavulugjimene8869 3 ай бұрын

Atbilde nepareiza! a = b = 9,0586215 ☺

@oahuhawaii2141 3 ай бұрын

@pavulugjimene8869: Your value is a little bit off. And there are 3 more solutions. There are 2 irrational solutions with a = b . They're the roots from the quadratic equation for a² - a = 73 . (1 + √293)/2 ≈ 9.05862138431184... (1 - √293)/2) ≈ -8.05862138431184... There are 2 integer solutions with a + b = -1 . They're the roots from the quadratic equation for a² + a = 72 . (a, b) = (8, -9), (-9, 8) { KZbin moderator: There's nothing wrong with these higher-precision numbers. }

@godot-lee 5 ай бұрын

a = 8, b = -9 or a = -9, b = 8

@thomasharding1838 4 ай бұрын

Or you could just say, one equals -9 and the other equals 8.

@tomtke7351 6 ай бұрын

a^2-b = b^2-a = 73 a^2-b^2=b-a (a-b)(a+b)=(b-a) (a+b)=(b-a)/(a-b) a+b = -1 eq.1 a=-b-1 a^2-b=73. eq.2 (-b-1)^2-b =73 b^2+2b+1-b=73 b^2+b+1=73 b^2+b-72=0. eq.3 (b+9)(b-8)=0 sol.1 b=-9 sol.2 b=8 from eq.1 a+b=-1 if b=-9 a-9=-1 a=8 if b=8 a+8=-1 a=-9 solutions (a,b)= (8,-9) = (-9,8) VERIFY a^2-b=?73 (8,-9)? 64--9=?73 73=❤73✔️ (-9,8) 81-8=?73 73=❤73✔️ b^2-a has identical results Conclude solutions (a,b)= (8,-9) = (-9,8)

@garrettvanmeter5831 6 ай бұрын

Write as (a,b) = (8,-9) or (a,b) =(-9,8 ). What the solutions as written say is (8,-9) =(-9,8) which is incorrect. Also only One solution is needed Since by symmetry if (8,-9) is a solution then so is (-9,8).

@賴瓊川 3 ай бұрын

The way is same with mine. Your solution is easier!

@LogicRk 6 ай бұрын

Superb ❤

@sadraouimohamed4539 3 ай бұрын

Gooooood

@Frankkusi-se2rf 5 ай бұрын

Please there is a problem in the factorization

@duncan-rmi 6 ай бұрын

I did it in my head. but what sort of pen is that? do they do them in other colours?

@ganeshdas3174 Ай бұрын

a = - 9 and b = 8 are satisfying the given conditions

@Leungwinnie_ 3 ай бұрын

❤very good lecture May I suggest a method: a+b=-1and ab=-72 Then we can formulate a quadratic equation: x^2+x-72=0. By quadratic formula, the two roots are 8 and -9 respectively😊

@oahuhawaii2141 3 ай бұрын

Here's a faster and cleaner solution. a² - b = 73 { E1 } b² - a = 73 { E2 } a² - b² + a - b = 0 { E1 - E2 } (a - b)*(a + b + 1) = 0 b = a , b = -a - 1 Substitute b = a in E1: a² - a = 73 a² - a - 73 = 0 a = b = (1 ± √293)/2 Substitute b = -(a + 1) in E1: a² + a + 1 = 73 a² + a - 72 = (a + 9)*(a - 8) = 0 a = (-1 ± 17)/2 = 8, -9 b = -(a + 1) = (-1 ∓ 17)/2 = -9, 8 The solution to the thumbnail: (a, b) = (8, -9), (-9, 8), ((1 ± √293)/2, (1 ± √293)/2) The video adds a requirement that a ≠ b, so: (a, b) = (8, -9), (-9, 8)

@vijay00071 2 ай бұрын

My first thought was it's either -8 and 9 or 8 and -9, you substitute em and you get your answer

@陈庶-i3h Ай бұрын

so easy. a^2-b=b^-a， a^2-b^2= b-a, (a+b)(a-b) = -(a-b) . a-b≠0 . a+b=-1. so b^2+b+1=73 ,b(b+1)=72 ,b=8, a=-9

@Tiberius88 6 ай бұрын

Long winded. It doesn't take 13 mins to solve this!

@YAWTon 6 ай бұрын

You can halve the time by playing the clip at double speed...😊

@oahuhawaii2141 3 ай бұрын

@YAWTon: Yes, I Press &:Hold on an empty spot on the screen, and the video player goes to double speed. If I double-tap, it skips ahead by 10 secs. I do both a lot. On a PC, I can press a digit to jump to a multiple of 10% of the video: "5" jumps to the midpoint. The right arrow key skips ahead by 5 secs.

@無姓無名-o9e 3 ай бұрын

a^2-b=73 b^2-a=73 a^2-b=b^2-a a^2-b^2=-a+b (a+b)(a-b)=-(a-b) a+b=-1 a=-(b+1) (-(b+1))^2-b=73 b^2+2b+1-73=0 b^2+b-72=0 (b+9)(b-8)=0 b=-9,8 (-9)^2-a=73 81-a=73 a=8 8^2-a=73 64-a=73 a=-9 Answer a=-9,b=8 or a=8,b=-9

@lu.cicerone.cavalheiro 3 ай бұрын

Lots of uneeded steps to solve that

@ИльяИванов-к4е 5 ай бұрын

В первую секунду я подумал, что a=b, и во вторую секунду меня обломали)

@oahuhawaii2141 3 ай бұрын

The solution to the thumbnail: (a, b) = (8, -9), (-9, 8), ((1 ± √293)/2, (1 ± √293)/2) The video adds a requirement that a ≠ b, so: (a, b) = (8, -9), (-9, 8)

@Sevochka11Azim 5 ай бұрын

Очень интересно. Спасибо

@Sevochka11Azim 5 ай бұрын

Хотя есть и более простые методы решения, но эта игра с формулами мне понравилась.

@learncommunolizer 5 ай бұрын

Your welcome

@EstebanChacon-e3e Ай бұрын

En la presentación del problema no dice que a es diferente de b

@Samk-nv6wp 3 ай бұрын

The comments are proof that mathematics is magical!

@als2cents679 6 ай бұрын

In your solution validation you only plugged values in one of the simultaneous equation. You really need to plug into both of the original simultaneous equations to validate that the solution is valid.

@oahuhawaii2141 3 ай бұрын

Yes. But since a and b are interchangeable in the symmetric system of equations, he only needs to do one complete set.

@als2cents679 3 ай бұрын

@@oahuhawaii2141 Yes, 1 set but both equations.

@oahuhawaii2141 3 ай бұрын

@als2cents679: Technically, he did the equivalent because he used both solutions for equation 1. The 1st solution in equation 1 is the 2nd solution in equation 2; likewise, the 1st solution in equation 2 is the 2nd solution in equation 1. That's due to the symmetry of the variables in the equations. It'll be clearer if he checked the 1st solution in both equations, and then say checking solution 2 is like checking solution 1 with the equations swapped.

@als2cents679 3 ай бұрын

@@oahuhawaii2141 Ok, got what you are saying. But yes it was a confusing way to explain stuff.

@als2cents679 3 ай бұрын

@@oahuhawaii2141 You see, if someone saw this solution and then applied to another problem where there was only one solution or problem was not symmetrical, then they might incorrectly assume it is enough just to check the answer with one of the simultaneous equation. On the other hand if you checked 1 solution fully and then mentioned that there is no need to check the other due to symmetry, then you won't have this problem.

@CharlesChen-el4ot 4 ай бұрын

(a-b)*(a+b -1) =0 a =b or a =1 -b a^2 - a = 73 a^2 - a - 73 =0

@oahuhawaii2141 3 ай бұрын

You have a wrong sign. (a - b)*(a + b + 1) = 0 The solution to the thumbnail: (a, b) = (8, -9), (-9, 8), ((1 ± √293)/2, (1 ± √293)/2) The video adds a requirement that a ≠ b, so: (a, b) = (8, -9), (-9, 8)

@Ananwerapun Ай бұрын

Two parabola graph

@2010chansir 4 ай бұрын

PURPOSELY PROLONG THE SOLUTION TO MAKE A VIDEO ?

@rajibhossain8311 3 ай бұрын

Correctly said. Just by look at the equation one and see the solution

@aqlimursadin5948 5 ай бұрын

It took me 20" to solve while eating my pepperoni pizza.

@oahuhawaii2141 3 ай бұрын

At 20", that's a big pizza. Pizza Hut's Personal Pan Pizza is 6", so you've eaten 11⅑ times that. If you can handle 70+ hotdogs, I'll see you on TV next July 4th.

@oahuhawaii2141 3 ай бұрын

a² - b = 73 { E1 } b² - a = 73 { E2 } a² - b² + a - b = 0 { E1 - E2 } (a - b)*(a + b + 1) = 0 b = a , b = -a - 1 Substitute b = a in E1: a² - a - 73 = 0 a = b = (1 ± √293)/2 Substitute b = -(a + 1) in E1: a² + a + 1 = 73 a² + a - 72 = 0 (a + 9)*(a - 8) = 0 a = -9, 8 b = -(a + 1) = 8, -9 The solution to the thumbnail: (a, b) = (-9, 8), (8, -9), ((1 ± √293)/2, (1 ± √293)/2) The video adds a requirement that a ≠ b, so: (a, b) = (-9, 8), (8, -9)

@dellagobaikal8205 5 ай бұрын

The proposed solution is too much complicated. Below proposed by G is more technical.

@lugracias 6 ай бұрын

👍🌼

@TWJRPGGamming 5 ай бұрын

a=-9 b=8

@laogui2425 2 ай бұрын

So long winded! a^2-b^2 = b-a and since b!=a can divide by a-b a+b=-1 a^2-b=a^2+a+1=73 a^2+a-72=0, (a-8)(a+9)=0 a=8 or a=-9, b= etc. 8^2+9=9^2-8=73

@МихайлоЛовга-ф1ц 5 ай бұрын

а=-9, b=8

@MuhammadSharjeelAnsari-s8w 6 ай бұрын

🎉

@totorowu5034 5 ай бұрын

應該有個環節錯誤不然最後不必這樣證明是嗎

@эльдар-в4п 6 ай бұрын

81-8=73 64+9=73

@Zbigniew-b3u 6 ай бұрын

:))))))))) Jak można tak rozwlekać proste zadanie ?!!! było wstawić a=-b-1 do drugiego równania i wynik gotowy z prostego równania kwadratowego b=-9 ; 8 a=8 ; -9

@YAWTon 6 ай бұрын

The same problem with the same solution was uploaded to YT a short while ago. I think this guy just made a copy of that clip... terribly boring, and bad from a didactic point of view.

@Qoow8e1deDgikQ9m3ZG 3 ай бұрын

(x^2-73)^2-x-73=0 (x-8)(x+9)(x^2-x-73)=0 hahahaha🤣🤣🤣🤣🤣🤣🤣🤣

@martamara1979 5 ай бұрын

Muy largo, y tiene más cosas que una piñata.

@vanessawelles4760 6 ай бұрын

easy to guess the solution by saying what number squared is is slightly greater than 73. result -9 and 8

@salvatorechiocca2375 6 ай бұрын

@oahuhawaii2141 3 ай бұрын

But you're assuming the solutions are always integers. They can be other types of real numbers, or even complex. The other 2 solutions are a = b = (1 ± √293)/2 ; they work for the thumbnail, before he added the a ≠ b requirement to his video.

@lisandro73 6 ай бұрын

There are two more real, when a and b are irrational

@oahuhawaii2141 3 ай бұрын

Yes, his thumbnail doesn't have the a ≠ b restriction.

@uouosxlyz9979 3 ай бұрын

Nasil olimpiyat sorusu bu.Benim gibi birinin kafadan yaklaşık bir dakikada çözdüğü sorunun olimpiyat sorusu olması garip.

@эльдар-в4п 5 ай бұрын

а=-9 в=8 !!!

@Андрей-я7и1с 6 ай бұрын

Всё конечно красиво, но зачем так сильно раскладывать и расписывать очевидные вещи?

@oahuhawaii2141 3 ай бұрын

@эльдар-в4п 6 ай бұрын

а=9 б=-8 !!!

@oahuhawaii2141 3 ай бұрын

You have wrong signs. The solution to the thumbnail: (a, b) = (8, -9), (-9, 8), ((1 ± √293)/2, (1 ± √293)/2) The video adds a requirement that a ≠ b, so: (a, b) = (8, -9), (-9, 8)