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.

Comments 30 comments

p l patel 4 years ago


urmi 4 years ago

not bad

Tvrtko 4 years ago

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

Caleb DRC profile image

Caleb DRC 4 years ago

This is fantastic; I had no idea, and I have never seen it explained in this way. Great job!

basavaraj 3 years ago

in the same way, do u know (a-b)^2

Abir 3 years ago

nice prooof

aysha 3 years ago

really helpful proof

ashish chandran 3 years ago

explainted very well.

najwa 3 years ago

very useful

messi jan 3 years ago

very good

mathsmaster profile image

mathsmaster 3 years ago from Islamabad, Pakistan

coooool :D

Loved it :)


zoya khan 3 years ago

Not Bad but it can be shorterand thanks for giving me answer . THANK YOU HUBPAGE.COM.THANK YOU VERY MUCH

rashmi jain 3 years ago


SACHIN 2 years ago

wwhat awomderfull

sally 2 years ago

tnx was helpful

aanchal 2 years ago


varun 2 years ago

How d formula z used in daily life

yosef 2 years ago

it is very useful for our learnig

sathya prathap 21 months ago

Thanks. Very useful

satyam 17 months ago

Tnx useful for me

Aryan aman 16 months ago

Spcl thnx

Pravallika 14 months ago


Angad chauhan 13 months ago


11 months ago

Good job

Aarthi 8 months ago

very useful

abhishek kumar singh 6 months ago

good job good site

ASif ali lone 6 months ago

Very useful and interesting

kaushik raj 2 months ago

can you plz proo another formula

pasha 8 weeks ago

How many a and how many b in a+bwhole square..?

sumati joshi 2 weeks ago

thanks for this

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