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

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[@giannaleoci2328]

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]

Smarter method (gave it a like)

[@garrettvanmeter5831]

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

[@is7728]

@@garrettvanmeter5831 Agree

[@ЛидийКлещельский-ь3х]

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]

Recall the condition that a is not equal to b.

[@robertliu3176]

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.

[@margotconard7644]

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]

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]

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

[@丁爸爸-w3f]

@@爷爷-m2p 你认为谁能看懂汉字？

[@oahuhawaii2141]

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]

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

[@evgenysapotnitsky8234]

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]

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

[@evbdevy352]

Потратил не более 5 мин.На Олимп.ге тянет

[@НадеждаМаксимович-ф3е]

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

[@pavulugjimene8869]

Atbilde nepareiza! a = b = 9,0586215 ☺

[@oahuhawaii2141]

@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]

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

[@thomasharding1838]

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

[@tomtke7351]

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]

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]

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

[@稲次将人]

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)

[@ИльяИванов-к4е]

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

[@oahuhawaii2141]

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)

[@mzallocc]

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)

[@Leungwinnie_]

❤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]

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)

[@johnplong3644]

You MUST well organize and neat when doing this problem

[@LogicRk]

Superb ❤

[@sadraouimohamed4539]

Gooooood

[@oahuhawaii2141]

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)

[@Samk-nv6wp]

The comments are proof that mathematics is magical!

[@2010chansir]

PURPOSELY PROLONG THE SOLUTION TO MAKE A VIDEO ?

[@rajibhossain8311]

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

[@soniamariadasilveira7003]

I loved this problem!

[@Frankkusi-se2rf]

Please there is a problem in the factorization

[@CharlesChen-el4ot]

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

[@oahuhawaii2141]

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)

[@darliney]

암산으로 풀었다 수가 복잡해지면 불가능 하겠지만 한자리수 답이라서 가능

[@als2cents679]

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]

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

[@als2cents679]

@@oahuhawaii2141 Yes, 1 set but both equations.

[@oahuhawaii2141]

@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]

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

[@als2cents679]

@@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.

[@duncan-rmi]

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

[@mohsenahmadi5575]

🎉

[@dellagobaikal8205]

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

[@Tiberius88]

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

[@YAWTon]

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

[@oahuhawaii2141]

@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]

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

[@Sevochka11Azim]

Очень интересно. Спасибо

[@Sevochka11Azim]

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

[@learncommunolizer]

Your welcome

[@karvindmishra09]

a=-9, b=8

[@Zbigniew-b3u]

:))))))))) 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]

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.

[@lugracias]

👍🌼

[@totorowu5034]

應該有個環節錯誤 不然最後不必這樣證明 是嗎

[@lisandro73]

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

[@oahuhawaii2141]

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

[@TWJRPGGamming]

a=-9 b=8

[@aqlimursadin5948]

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

[@oahuhawaii2141]

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.

[@uouosxlyz9979]

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

[@martamara1979]

Muy largo, y tiene más cosas que una piñata.

[@vanessawelles4760]

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

[@salvatorechiocca2375]

A

[@oahuhawaii2141]

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.

[@эльдар-в4п]

81-8=73 64+9=73

[@evbdevy352]

I have solved by other way.Plused (1) and (2)

[@эльдар-в4п]

а=-9 в=8 !!!

[@Qoow8e1deDgikQ9m3ZG]

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

[@Андрей-я7и1с]

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

[@oahuhawaii2141]

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)

[@arshadshaikh5905]

a = b = 9.05863

[@oahuhawaii2141]

a = b ≈ 9.0586213843118..., -8.058621384311...

[@oahuhawaii2141]

a = b ≈ 9.0586213843118..., -8.058621384311... KZbin moderator: there's nothing wrong with these higher precision numbers.

[@эльдар-в4п]

а=9 б=-8 !!!

[@oahuhawaii2141]

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)

[@soshakobyan3123]

Olympiad problem? I think this is a joke.

[@oahuhawaii2141]

Breakdancing as an Olympic sport is a joke, too.

[@MuhammadSharjeelAnsari-s8w]

🎉

[@МихайлоЛовга-ф1ц]

а=-9, b=8

[@吳昶霖]

a=8 b=-9