Math Olympiad | A Nice Algebra Problem | How to solve for "a" and "b" in this Problem?
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- Опубликовано: 8 июл 2024
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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)
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).
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
Smarter method (gave it a like)
Again, only one solution is needed since you can appeal to symmetry to get the other
@@garrettvanmeter5831 Agree
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.
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。
It took me around 10 sec to find the solution. Glory to Ukraine.
В первую секунду я подумал, что a=b, и во вторую секунду меня обломали)
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 to, Lidiy
Recall the condition that a is not equal to b.
Superb ❤
Please there is a problem in the factorization
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.
👍🌼
I did it in my head. but what sort of pen is that? do they do them in other colours?
🎉
Очень интересно. Спасибо
Хотя есть и более простые методы решения, но эта игра с формулами мне понравилась.
Your welcome
The proposed solution is too much complicated.
Below proposed by G is more technical.
a = b = 9.05863
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))
Я тоже решил именно так,отнял от(1) (2).Получил такое же кв.урав-е.
Потратил не более 5 мин.На Олимп.ге тянет
a = 8, b = -9 or a = -9, b = 8
I loved this problem!
a=-9
b=8
81-8=73 64+9=73
а=-9, b=8
:))))))))) 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
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.
а=9 б=-8 !!!
There are two more real, when a and b are irrational
easy to guess the solution by saying what number squared is is slightly greater than 73. result -9 and 8
A
a=8 b=-9
Long winded. It doesn't take 13 mins to solve this!
You can halve the time by playing the clip at double speed...😊
Всё конечно красиво, но зачем так сильно раскладывать и расписывать очевидные вещи?
I have solved by other way.Plused (1) and (2)
Olympiad problem? I think this is a joke.