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Why (a+b) 2 = a2+b2+2ab ?

Updated on June 01, 2015

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.


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    • profile image

      p l patel 4 years ago


    • profile image

      urmi 4 years ago

      not bad

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      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!

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      basavaraj 4 years ago

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

    • profile image

      Abir 4 years ago

      nice prooof

    • profile image

      aysha 4 years ago

      really helpful proof

    • profile image

      ashish chandran 4 years ago

      explainted very well.

    • profile image

      najwa 3 years ago

      very useful

    • profile image

      messi jan 3 years ago

      very good

    • mathsmaster profile image

      Hassan Shahbaz 3 years ago from Islamabad, Pakistan

      coooool :D

      Loved it :)


    • profile image

      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

    • profile image

      rashmi jain 3 years ago


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      SACHIN 3 years ago

      wwhat awomderfull

    • profile image

      sally 2 years ago

      tnx was helpful

    • profile image

      aanchal 2 years ago


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      varun 2 years ago

      How d formula z used in daily life

    • profile image

      yosef 2 years ago

      it is very useful for our learnig

    • profile image

      sathya prathap 2 years ago

      Thanks. Very useful

    • profile image

      satyam 22 months ago

      Tnx useful for me

    • profile image

      Aryan aman 21 months ago

      Spcl thnx

    • profile image

      Pravallika 19 months ago


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      Angad chauhan 18 months ago


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      17 months ago

      Good job

    • profile image

      Aarthi 13 months ago

      very useful

    • profile image

      abhishek kumar singh 11 months ago

      good job good site

    • profile image

      ASif ali lone 11 months ago

      Very useful and interesting

    • profile image

      kaushik raj 8 months ago

      can you plz proo another formula

    • profile image

      pasha 7 months ago

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

    • profile image

      sumati joshi 5 months ago

      thanks for this

    • profile image

      Sourab 4 months ago

      I really liked the proof and I never knew that it can be proved in this way tooooo......

    • profile image

      Hazique 3 months ago

      Thank you sir

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      Likith 2 months ago

      Nice explanation it is easy to learn

    • profile image

      samyam 10 days ago


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