Why (a+b) 2 = a2+b2+2ab ?
Why (a+b) 2 = a2+b2+2ab ?
Ever wondered how was the above formula derived?
Probably the answer would be yes and is simple. Everybody knows it and when you multiply (a+b) with (a+b) you will get a plus b whole square.
(a+b) * (a+b) = a2 +ab + ba + b2 = a2 + 2ab + b2
But how did this equation a plus b whole square became generalized.
Let’s prove this formula geometrically.( Please refer to the pictures on the side)
- Consider a line segment.
- Consider any arbitrary point on the line segment and name the first part as ‘a’ and the second part as ‘b’. Please refer to fig a.
- So the length of the line segment in fig a is now (a+b).
- Now, let’s draw a square having length (a+b). Please refer to fig b.
- Let’s extend the arbitrary point to other sides of the square and draw lines joining the points on the opposite side. Please refer to fib b.
- As we see, the square has been divided into four parts (1,2,3,4) as seen in fig b.
- The next step is to calculate the area of the square having length (a+b).
- As per fig b , to calculate the area of the square : we need to calculate the area's of parts 1,2,3,4 and sum up.
- Calculation : Please refer to fig c.
Area of part 1 :
Part 1 is a square of length a.
Therefore area of part 1 = a2 ---------------------------- (i)
Area of part 2 :
Part 2 is a rectangle of length : b and width : a
Therefore area of part 2 = length * breadth = ba -------------------------(ii)
Area of part 3:
Part 3 is a rectangle of length: b and width : a
Therefore area of part 3 = length * breadth = ba --------------------------(iii)
Area of part 4:
Part 4 is a square of length : b
Therefore area of part 4 = b2 ----------------------------(iv)
So, Area of square of length (a+b) = (a+b)2 = (i) + (ii) + (iii) + (iv)
Therefore :
(a+b)2 = a2 + ba + ba +b2
i.e. (a+b)2 = a2 + 2ab + b2
Hence Proved.
This simple formula is also used in proving The Pythagoras Theorem.Pythagoras Theorem is one of the first proof in Mathematics.
In my view, in mathematics when a generalized formula has been framed there will be a proof to prove and and this is my small effort to exhibit one of the proof's.
Will definitely come up with some more.
Also please find the video proof.
Questions & Answers
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Comments
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What is the solution of (a+b)2=
Thanks a lot for help
I am not satisfied with that proof. Any other please!
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Answer in hindi
Actually good but as well as now its common.. way to prove but it will b more better if u find out some more way 2 prove ,becoz that may give u more greetings from everyone
Then also u tried ur best to help us so ..good job keep it up
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Mathematics =3 semester Question and Answer send me please VVI Question
How a2B 2 +aB2
A+b =a2*bxv
Tens
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Could u give me this kind of geometrical explanation for (a+b)³.?
Very nice and helpful to me thanx for it..........
Very Good.
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Thanks for this but can you give some questions related to this and its answer .please
hello bawa! today I really found out the mystry of the Universe.
Good, like it
Very help ful
thakyou
Nice explanation it is easy to learn
Thank you sir
I really liked the proof and I never knew that it can be proved in this way tooooo......
thanks for this
How many a and how many b in a+bwhole square..?
can you plz proo another formula
Very useful and interesting
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(a-b)2=a2+b2-2ab.
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How d formula z used in daily life
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(y)
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really helpful proof
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in the same way, do u know (a-b)^2
in the same way, do u know (a-b)^2
This is fantastic; I had no idea, and I have never seen it explained in this way. Great job!
Actually, it is proved by using Pascal's triangle, with the general formula: (a+b)^n = (n 0) a^n * b^0 + (n 1) a^n-1 * b^1 + ..... + (n n-1) a^1 b^n-1 + (n n) a^0 * b^n-1
Or shorter (a+b)^n = n Σ k=0 (n k) a^n-k * b^k
not bad
good
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