A nice algebra problem ।। maths olympiad ।।
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- เผยแพร่เมื่อ 5 ก.ย. 2024
- A nice algebra problem ।। maths olympiad ।। #algebra #mathslearning @mathscuriosity494
Nice Algebra simplification।। ।। maths olympiad question @mathscuriosity494
if, a2+b²=6ab then find the value of a+b/a-b
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2
√2
Applying componendo and dividendo roles will make this much easier
Radhey radhey bhaiya ji like dn 🙏
Square (a+b)/(a-b) and get the square root of the answer. (a+b)/(a-b) all squared= a^2 + b^2 + 2ab / ( a^2 + b^2 - 2ab) = 8ab/ 4ab = 2 then the square root of (a+b)/ (a-b)= plus or minus sqroot of 2
It can be shown that -√2 is also a valid answer.
The required expression =(a+b)/(a-b)
Let it be equal to u
i.e., u=(a+b)/(a-b)
Squaring both sides, we get: u²=(a²+2ab+b²)/(a²-2ab+b²) -- (i)
But it's given that a²+b²=6ab
Using in (i), we get:
u²=(6ab+2ab)/(6ab-2ab)=8ab/4ab=2
Thus, u=±√2
The problem has 2 answers.
You can convince yourselves that -√2 is also a valid answer by choosing a and b values as follows:
a=3-2√2; b=1 -- (ii)
a²+b²=(3-2√2)²+1²=9-12√2+(2√2)²+1=9-12√2+8+1=18-12√2
6ab=6.(3-2√2).1=18-12√2
Hence values given in (ii) clearly satisfy a²+b²=6ab
Now, let's find the value of (a+b)/(a-b)
= [(3-2√2)+1]/[(3-2√2)-1]=(4-2√2)/(2-2√2)=-√2
Hence it's seen that -√2 is also an answer.
Therefore, the problem has 2 answers: i.e., (a+b)/(a-b)=±√2
i solved it in 15sec
Bahut badhiya tarike se solve kiye ho , samajhaya bahut acche se 👌
Lk dn 👍 runing h bhai
(a + b) ^2 = 8 a b
( a - b) ^2 = 4 a b
Hereby ( a + b) ^2 / ( a - b) ^2 = 2
( a + b) /( a - b) = √2, - √2
Nice solution. However I do like to give another method which is short.
a^2 + b^2 =6ab
Dividing both sides by 2ab we may get
(a^2 + b^2 )/2ab = 3
By C & D
(a +b) ^2/(a-b) ^2=4/2=2
Taking square root of both sides we may get
(a+b)/(a -b) = +√2 , - √2
Comment please.
Radhe radhe bhai ji 🙏🙏👌👌👍👍babita 🎉🎉
It is very nice solution. But, it could be that there is another solution, which is minus square root of 2. Please check it out. Thanks!!!
This can be done in much shorter step
Very nice shearing bhaiya ji 🎉🎉🎉🎉🎉🎉 God bless you always
Thank you so much
Very nice 👌🥰
Thanks 🤗
Nice sharing ❤
good day. Take care both a and b not equal to 0 and a not equal to b. best regards
🎉🎉🎉good morning🙏🙏🙏🙏 lk 6:30
Nice sharing video 🎉 like 👌🏻👍
Thanks for visiting
13lk very nice sharing ❤😊😊😊
Thank you so much sister 🙏🏻🙏🏻
wow so wonderful sir
Very nice sharing like done
बहुत बढ़िया सबसे बढ़िया ढंग समझाने का 6:35
Very nice🎉🎉🎉🎉
+or-
a²+b²=6ab
(a+b)/(a-b)?
(a+b)²-2ab=6ab
(a+b)²=8ab
|a+b|=√(8ab)
a+b=±√(8ab)
a+b=±2√(2ab)
(a-b)²+2ab=6ab
(a-b)²=4ab
|a-b|=√(4ab)
a-b=±√(4ab)
a-b=±2√(ab)
(a+b)/(a-b)=±√2 ❤❤
❤❤❤❤🎉🎉🎉🎉🎉
Actually, there could be two answers unless there are constraints on a and b: e.g.
(a+b)^2/(a-b)^2 = (8ab)/(4ab) = 2. Therefore, taking square roots of both sides, (a+b)/(a-b) = (+/-). Dr. Ajit Thakur (USA).
Hi, thanks
I think there is a shorter way.
a² + b² + 2ab = (a + b)² = 8ab
a² + b² - 2ab = (a - b)² = 4ab
So (a + b)²/(a - b)² = 8ab/4ab = 2
which means that
(a + b)/(a - b) = ±sqrt(2)
(a+b)^2=a^2+b^2+2ab....(1)
(a-b)^2=a^2+b^2-2ab....(2)
(1) devided (2)
((a+b)/(a-b))^2
= (a^2+b^2+2ab)/(a^2+b^2-2ab)......(3)
let: a^2+b^2=6ab
Eq. (3)
(a+b)/(a-b))^2
= (6ab+2ab)/(6ab-2ab)
=8ab/4ab
=2..........(4).
finaly:
Eq.4
(a+b)/(a-b))^2 =2
(a+b)/(a-b)=√2 ....end
Beautiful explanation
Nice sharing
Great Upload
Very nice ❤
Much easy way if you apply formula a plus b whole square minus a minus b whole square which is equal to 4ab
18 like good 💯 bahut acha se sawal solve krte h
Thank you so much sister 🙏🏻🙏🏻
Instead of extending the steps it can be done by adding 2ab in both sides and subtracting 2ab from both sides.
19 like done bahut badhiya very nice sharing ❤❤❤❤❤
Bahut hi useful video bhai 👌 radhe radhe bhai 🙏
Waching here ✅👍👌
13 like🙏🥰 on sir
Many many thanks
😊 6:35
Another is minus square root 2.
Lk dn ❤❤❤
Very good explanation sir Good afternoon 🙏
Very useful video 😊
sq.root 2
Very nice 🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏
Bye bye
Very nice shering bhai 👌🏽 👍🏽 🙏🙏🙏🙏
❤❤❤❤
Sari video bahut useful hoti hai❤❤
Thank you so much sister 🙏🏻🙏🏻
186Ld❤❤
The other answer may be -√2 also if b > a and b-a = negative.
Lk🎉
So beautiful 😍😍
Thank you! 😊
Very lengthy process
We solve it as
a²+b²= 6ab ⟹ (a+b)² =8ab
a²+b² =6ab ⟹(a -b)² =4ab
Thus
(a+b)/(a-b) =√{ (a+b)²/(a-b)²}= √{(8ab)²/(4ab)²}
Which is nothing but equal to √2
#naweenraaj
I have given a short process and it bears two answers
Very nice shering bhai Thank-you for sharing 👌🏽
Radhey Radhey bhaiya 🙏🙋♀️
Very nice 🎉
too lengthy
jai shree RAM bhai so Beautiful sharing Nice video ji
Like done
very informative vlog for students
Thanks a lot
Nice ❤❤❤❤❤like ❤❤❤❤❤
Sr ji 🙏 🤗❤
Lk🙏🙏🙏
Nice
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Thank you so much sister 🙏🏻
Very nice
Thanks
Your solution is so difficult,
Super solution guru ji 🙏
Решение неверно, в условии задачи ничего не сказано про а и b
А если аb или ( - sqrt 2) если а
Like done watching
from given expn: (a+b)^2=8ab and (a-b)^2=4ab=>(a+b)/(a-b)=2^1/2.ans
H lλύση είναι λάθος α>β???
The procedure is very rough. Solving procedure is very complex. There is an easiest way to solve this problem. As a mathematician I don't like your procedure. Are you a mathematician? I think you are not a professional teacher of mathematics.
U could reduce one or two step.(a+b)^2 =8ab, (a-b)^2 =4ab . Hence it's division is 2, then a+b/a-b = root over 2.
This can be done in much shorter step
√2
Very nice sharing sir 🙏
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Thank you 👍
Lk🎉
very nice sir
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Thanks for visiting
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