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

Updated on June 1, 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 5 years ago

      good

    • profile image

      urmi 4 years ago

      not bad

    • profile image

      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!

    • profile image

      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 4 years ago

      very useful

    • profile image

      messi jan 4 years ago

      very good

    • mathsmaster profile image

      Hassan Shahbaz 4 years ago from Islamabad, Pakistan

      coooool :D

      Loved it :)

      (y)

    • 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

      thanx

    • profile image

      SACHIN 3 years ago

      wwhat awomderfull

    • profile image

      sally 2 years ago

      tnx was helpful

    • profile image

      aanchal 2 years ago

      thankyou

    • profile image

      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 2 years ago

      Tnx useful for me

    • profile image

      Aryan aman 2 years ago

      Spcl thnx

    • profile image

      Pravallika 23 months ago

      Good

    • profile image

      Angad chauhan 22 months ago

      (a-b)2=a2+b2-2ab.

    • profile image

      20 months ago

      Good job

    • profile image

      Aarthi 17 months ago

      very useful

    • profile image

      abhishek kumar singh 15 months ago

      good job good site

    • profile image

      ASif ali lone 15 months ago

      Very useful and interesting

    • profile image

      kaushik raj 11 months ago

      can you plz proo another formula

    • profile image

      pasha 10 months ago

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

    • profile image

      sumati joshi 9 months ago

      thanks for this

    • profile image

      Sourab 8 months ago

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

    • profile image

      Hazique 7 months ago

      Thank you sir

    • profile image

      Likith 6 months ago

      Nice explanation it is easy to learn

    • profile image

      samyam 4 months ago

      thakyou

    • profile image

      pranav Aryan 3 months ago

      Very help ful

    • profile image

      Suraj 3 months ago

      Good, like it

    • profile image

      tutku 7 weeks ago

      hello bawa! today I really found out the mystry of the Universe.

    • profile image

      Shiny 7 weeks ago

      Thanks for this but can you give some questions related to this and its answer .please

    • profile image

      sunil basnet 6 weeks ago

      good

    • profile image

      Saumya 5 weeks ago

      Thanks for the article . It was very helpful to me

    • profile image

      Rahul raj 2 weeks ago

      How are you friend

    • profile image

      Sam 11 days ago

      Nice sir...........""

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