A Nice Algebra Problem | Math Olympiad | Find a and b?

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Find the value of a and b?
How to solve a² + 2ab + b = 22

In this video, we'll show you How to Solve Math Olympiad Question A Nice Algebra Problem a² + 2ab + b = 22 in a clear , fast and easy way. Whether you are a student learning basics or a professtional looking to improve your skills, this video is for you. By the end of this video, you'll have a solid understanding of how to solve math olympiad exponential equations and be able to apply these skills to a variety of problems.
#matholympiad #maths #math #algebra

[davidseed]

Here is the quickest way.
Rearrange b= (22-a^2)/(2a+1)
Now scan through all the possible values of a and see what integer values of b occur until the calculated value of b is <1
a=1 b =21/3 =7
a=2 b=18/5 No
a=3 b=13/7 No
a=4 b=6/9 <1 so end of test
a=1, b=7 is the only solution.

[zawatsky]

(a+b)²=22-b²+b. b²>b, значит правая часть меньше 22. Ближайшие целые квадраты: 16, 9, 4, 1, 0. Подставляем 0. a=-b⇔ a<0 ∨ b<0 - вне области определения. 0<a=1-b, 1=22-b²+b⇔b²-b-21=0. D=1+4*21=85. Нет целых корней, идём дальше. 0<a=2-b, b²-b-22+4=b²-b-18. D=1+4*18=1+72=73. Та же история. Дальше. 0<a=3-b, b²-b-22+9=b²-b-13=0. D=1+4*13=53. Дальше. 0<a=4-b, b²-b-22+16=b²-b-6=0. D=1+4*6=25 - есть целый корень! b=(1±5)/2 - положительный 6/2=3. а=4-3=1>0. Единственный корень найден. Проверяем: 1+2*1*3+3=10 - мало. %) Попробуем наоборот: 9+2*1*3+1=9+6+1=16 - снова мало. Что же получается - совсем корней нет?🤔

[ManojkantSamal]

a=1, b=7 and vice versa...
Explain later...