Japanese | Can you solve this ? | A Nice Math Olympiad Algebra Problem

Рет қаралды 67,752

5 ай бұрын

Hello My Dear Family😍😍😍
I hope you all are well 🤗🤗🤗
If you like this video about
How to solve this math problem
please Like & Subscribe my channel as it helps me alot ，🙏🙏🙏🙏

Пікірлер: 40

@Berin.Jervin 10 күн бұрын

Nice problem. Good solution

@ms9035 5 ай бұрын

After reaching the following equation, it is better to do the rest of the operations in the following simple and easy way to avoid the complexity of the answer. ( a+b)^2 + 2( a+b) = 8 (a+ b )^2 +2 (a+b) + 1 = 8+1 = 9 ( ( a+b) +1 )^2 = 9 (a+b)+1 = 3 or ( a+b)+1 = -3 a+b-2 =0 or a+b+4 = 0 And then we will continue to write the answer in the same way as you have done. Thank you my best friend

@dmitrynikiforov8198 5 ай бұрын

Absolutely, so did I too.

@michelnkoghe7363 2 ай бұрын

After obtaining équations 1 and 2 as following Eq1: a^2 + a + b + ab = 3 Eq2: b^2 + a + b + ab = 5, Why not substrat them directly by doing Eq1 - Eq2? Thus a^2 - b^2 = 3 - 5 = - 2; (Eq3); (a + b)(a - b) = -2 = (-1).2 = (-2).1; the 1st couple ((a+b),(a+b)) brings from Eq3: a + b = -1 (Eq4) and a - b = 2 (Eq5); by adding them, we get 2a = 1; a = 1/2; wuth Eq5: 1/2 -b = 2;; b = 1/2 - 2 = -3/2 => 1st couple (a,b) = (1/2, -3/2); thé 2nd couple ((a+b),(a-b)) brings from Eq3: a + b = -2 (Eq6) and a - b = 1 (Eq7); by adding them, we get 2a = -1; a = -1/2; wuth Eq6: -1/2 + b = -2;; b = 1/2 - 2 = -3/2 => 2st couple (a,b) = (-1/2, -3/2); remembering that a = √(X - 1) and b = √(X + 1), Squaring both a and b whatever thé couples WE got brings this: X - 1 = 1/4 X + 1 = 9/4; Adding them brings 2X = 10/4 = 5/2; thus X= 5/4 >= 1. Thé solution IS thé same: X = 5/4

@sharatchandrasekhar2711 4 ай бұрын

Just substitute y=sqrt(x-1) and collect terms into the form sqrt(y^2 + 2) = (4 - y^2)/(1+y) - 1 Now square both sides and rearrange to get (y^2 + 2)(1 + y)^2 = (3 - y - y^2)^2 Expand and note that what appears to be a quartic actually degenerates to a quadratic with only one non-spurious root. The answer x=5/4 follows.

@gerhardb1227 4 ай бұрын

Thank you for your solution: Here another approach: substitute x = a^2 + 1 will result in a quadratic equation => 8a² +10a-7 = 0 a1 = 1/2; a2 = -7/4 a1 will be a valid solution = 1/2

@rosariobravo9165 4 ай бұрын

@gerhardb1227 Buenos días. Muy interesante su enfoque. Entiendo que sólo hace una sustitución de variable. Lo voy a intentar. Gracias.

@user-tl9bq7gd9v Ай бұрын

Lovely work!

@rober3072 4 ай бұрын

There is a mistake a² >= 0, then a >= 0 or a= 2 then b>= sqrt(2) or b= 1, then a >= 0 and b >= sqrt(2). The solution is OK, but the reasoning is incorrect

@knotwilg3596 Ай бұрын

Observe that the square of the inner terms equals 2 times the outer terms: (v(x-1)+v(x+1))² = 2x + 2v(x²-1) Set t = v(x-1)+v(x+1); (A) we get t + t²/2 = 4 or t²+2t-8=0 This quadratic equation has solutions t= -1 +/- 3 = 2 or -4. Since t is the sum of sqrts, it's positive and only t=2 is valid. Set u = v(x-1) then v(x+1) = v(u²+2) (A): 2 = u + v(u²+2) or 2-u = v(u²+2) square that, so (2-u)² = u²+2 so u²-4u+4 = u²+2 so -4u = -2 so u = 1/2 So x = 5/4

@MartinPerez-oz1nk 5 ай бұрын

THANKS PROFESOR !!!, VERY INTERESTING !!!!

@learncommunolizer 5 ай бұрын

You are welcome! Thank you very much!!

@jarikosonen4079 4 ай бұрын

At 4:01 if you add one on both sides, what will happen? (Numbers on right side getting bigger?) 17:46 Eq. 3, could be solved directly also backsubstituting both a and b. Key seems right substitution.

@rosariobravo9165 4 ай бұрын

Precioso ejercicio.

@learncommunolizer 4 ай бұрын

Thank you very much 👍👍

@MrUtubePete 2 ай бұрын

He sure took a loooong way to get there

@rezanader5770 Ай бұрын

Hi, at the beginning of solution the obtained domain is incomplete. You should also solve 4-x>=0. So the domain is 1=

@user-pv7jv3dc9s 4 ай бұрын

To be shortly solving, let's ander root x-1 + anger root x+1 =t, Than will be= t power 2+ 2time t+=8, » t+1 powe2 =9» t+1 =3 t=2 and etc.

@Pierre1O 5 ай бұрын

Nice solution! Note: Sqrt of any value is always positive, so a+b is always positive. No need for all these inequalities to prove a+b /= -4.

@yardenvokerol4253 4 ай бұрын

It's a very slippery slope to victory. if you take a wrong turn, you will find yourself in a dead end

@user-xc5os4ep3n Ай бұрын

Икс в квадрате минус один-это же (х-1)(х+1)😊

@captainteach007 3 ай бұрын

by the way, perfect square is not the same as difference of squares

@gruba4630 3 ай бұрын

Condition that square roots must be real numbers is not mentioned in the original problem, you introduced it yourself, so your solution is partial. Also, you spend too much time explaining what should be obvious for this math level. It would be nice to know which Japanese math olympiad it was, year and level. Otherwise, very nice presentation.

@ronaldnoll3247 5 ай бұрын

The result is correct... x=1.25

@nagarajahshiremagalore226 Ай бұрын

Pl write what you are telling

@Se-La-Vi 4 ай бұрын

Много лишних шагов с преобразованиями

@comdo777 5 ай бұрын

asnwer=3 isit

@user-it6fh7hy6t 5 ай бұрын

Профессор! Нормальный учитель распишет такое уравнение за 2 минуты,даже отвлекаясь на чай.

@user-im1hv7oc4w 2 ай бұрын

Mi fa male la testa seguirti.😢😢😢😢

@Nehezra2023 5 ай бұрын

In your solution, there is an error in solving (x+1)(×-1)>=0. You forgot (x+1)=0 AND B>=0) OR (A

@dmitrynikiforov8198 5 ай бұрын

Are you serious ? This is OLYMPIAD problem ? It has taken from me about 2 minutes to solve it. This is the problem for a university matriculant.

@user-it6fh7hy6t 5 ай бұрын

Какой же университет закончил наш профессор, если на такое примитивное уравнение он потратил 25 (!) минут?

@dmitrynikiforov8198 5 ай бұрын

@@user-it6fh7hy6t Сам удивляюсь :)

@l.w.paradis2108 5 ай бұрын

Well, so did I, but these are screening problems. People who don't solve a whole bunch of these really fast do not qualify.

@l.w.paradis2108 5 ай бұрын

All of these are screening problems from timed tests. The real problems are multi-step projects involving informal proofs.

@Billts 5 ай бұрын

This is penise exercise. I don't understand

@peterotto712 5 ай бұрын

Much too complicated - square the original equation twice

@schlingel0017 5 ай бұрын

It is more complicated like that and even the. you will still have some square roots left in the equation.

@bdfu4321 4 ай бұрын

无聊

@johnlee6304 5 ай бұрын

No need to write down that much