Math Olympiad | A Nice Algebra Problem | How to solve for "a" and "b" in this Problem?
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- เผยแพร่เมื่อ 8 ก.ค. 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.
Superb ❤
Please there is a problem in the factorization
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.
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.
👍🌼
В первую секунду я подумал, что a=b, и во вторую секунду меня обломали)
I did it in my head. but what sort of pen is that? do they do them in other colours?
I loved this problem!
🎉
a = b = 9.05863
The proposed solution is too much complicated.
Below proposed by G is more technical.
Очень интересно. Спасибо
Хотя есть и более простые методы решения, но эта игра с формулами мне понравилась.
Your welcome
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
а=-9, b=8
81-8=73 64+9=73
a=-9
b=8
There are two more real, when a and b are irrational
a=8 b=-9
:))))))))) 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 !!!
easy to guess the solution by saying what number squared is is slightly greater than 73. result -9 and 8
A
Всё конечно красиво, но зачем так сильно раскладывать и расписывать очевидные вещи?
Long winded. It doesn't take 13 mins to solve this!
You can halve the time by playing the clip at double speed...😊
I have solved by other way.Plused (1) and (2)
Olympiad problem? I think this is a joke.