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.





Questions & Answers

    Comments

      0 of 8192 characters used
      Post Comment
      • profile image

        Ramamani 

        9 days ago

        Can you give

        related questions

      • profile image

        Thanks 

        3 months ago

        Thanks

      • profile image

        Jaswanth 

        4 months ago

        Hi guys I'm u r new friend

      • profile image

        CHANDRU CHAN 

        6 months ago

        Nice

      • profile image

        kathie 

        6 months ago

        thanks

      • profile image

        PRADEEP singh 

        8 months ago

        It is actually better for me thanks for it

      • profile image

        deepak 

        8 months ago

        good

      • profile image

        Bidor Engti 

        8 months ago

        What is the solution of (a+b)2=

      • profile image

        Atul kumar pal 

        8 months ago

        Thanks a lot for help

      • profile image

        Janani 

        9 months ago

        I am not satisfied with that proof. Any other please!

      • profile image

        aparna 

        10 months ago

        nice

      • profile image

        shyam 

        10 months ago

        Good

      • profile image

        vinay pratap singh 

        11 months ago

        minded

      • profile image

        ishika 

        12 months ago

        very nice.........and helpful

      • profile image

        Pratima pandey 

        12 months ago

        Answer in hindi

      • profile image

        Aisha chowdhary 

        13 months ago

        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

      • profile image

        Uttam 

        14 months ago

        9771090194

      • profile image

        Mosaira Aaliya 

        15 months ago

        Diploma Electrical Engnnering

        Mathematics =3 semester Question and Answer send me please VVI Question

      • profile image

        Lan 

        16 months ago

        How a2B 2 +aB2

      • profile image

        parthasarathi 

        16 months ago

        A+b =a2*bxv

      • profile image

        Samay sahu 

        16 months ago

        Tens

      • profile image

        Nihal ahmad 

        16 months ago

        Vvvvv good

      • profile image

        Ignatius 

        17 months ago

        Could u give me this kind of geometrical explanation for (a+b)³.?

      • profile image

        Ritu 

        18 months ago

        Very nice and helpful to me thanx for it..........

      • profile image

        mr.krishna rai Gaurishriram (sirjam) 

        18 months ago

        Very Good.

      • profile image

        Sam 

        19 months ago

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

      • profile image

        Rahul raj 

        19 months ago

        How are you friend

      • profile image

        Saumya 

        20 months ago

        Thanks for the article . It was very helpful to me

      • profile image

        sunil basnet 

        20 months ago

        good

      • profile image

        Shiny 

        20 months ago

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

      • profile image

        tutku 

        20 months ago

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

      • profile image

        Suraj 

        22 months ago

        Good, like it

      • profile image

        pranav Aryan 

        22 months ago

        Very help ful

      • profile image

        samyam 

        23 months ago

        thakyou

      • profile image

        Likith 

        2 years ago

        Nice explanation it is easy to learn

      • profile image

        Hazique 

        2 years ago

        Thank you sir

      • profile image

        Sourab 

        2 years ago

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

      • profile image

        sumati joshi 

        2 years ago

        thanks for this

      • profile image

        pasha 

        2 years ago

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

      • profile image

        kaushik raj 

        2 years ago

        can you plz proo another formula

      • profile image

        ASif ali lone 

        2 years ago

        Very useful and interesting

      • profile image

        abhishek kumar singh 

        2 years ago

        good job good site

      • profile image

        Aarthi 

        3 years ago

        very useful

      • profile image

        3 years ago

        Good job

      • profile image

        Angad chauhan 

        3 years ago

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

      • profile image

        Pravallika 

        3 years ago

        Good

      • profile image

        Aryan aman 

        3 years ago

        Spcl thnx

      • profile image

        satyam 

        3 years ago

        Tnx useful for me

      • profile image

        DHANANJAY KUMAR 

        3 years ago

        THANKS IS VERY HELPFUL

      • profile image

        sathya prathap 

        4 years ago

        Thanks. Very useful

      • profile image

        yosef 

        4 years ago

        it is very useful for our learnig

      • profile image

        varun 

        4 years ago

        How d formula z used in daily life

      • profile image

        aanchal 

        4 years ago

        thankyou

      • profile image

        sally 

        4 years ago

        tnx was helpful

      • profile image

        SACHIN 

        4 years ago

        wwhat awomderfull

      • profile image

        SACHIN 

        4 years ago

        wwhat awomderfull

      • profile image

        rashmi jain 

        5 years ago

        thanx

      • profile image

        zoya khan 

        5 years ago

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

      • mathsmaster profile image

        Hassan Shahbaz 

        5 years ago from Islamabad, Pakistan

        coooool :D

        Loved it :)

        (y)

      • profile image

        messi jan 

        5 years ago

        very good

      • profile image

        najwa 

        5 years ago

        very useful

      • profile image

        ashish chandran 

        5 years ago

        explainted very well.

      • profile image

        aysha 

        6 years ago

        really helpful proof

      • profile image

        Abir 

        6 years ago

        nice prooof

      • profile image

        basavaraj 

        6 years ago

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

      • profile image

        basavaraj 

        6 years ago

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

      • Caleb DRC profile image

        Caleb DRC 

        6 years ago

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

      • profile image

        Tvrtko 

        6 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

      • profile image

        urmi 

        6 years ago

        not bad

      • profile image

        p l patel 

        6 years ago

        good

      working

      This website uses cookies

      As a user in the EEA, your approval is needed on a few things. To provide a better website experience, owlcation.com uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

      For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at: https://owlcation.com/privacy-policy#gdpr

      Show Details
      Necessary
      HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
      LoginThis is necessary to sign in to the HubPages Service.
      Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
      AkismetThis is used to detect comment spam. (Privacy Policy)
      HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
      HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
      Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
      CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
      Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the googleapis.com or gstatic.com domains, for performance and efficiency reasons. (Privacy Policy)
      Features
      Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
      Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
      Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
      Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
      Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
      VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
      PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
      Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
      MavenThis supports the Maven widget and search functionality. (Privacy Policy)
      Marketing
      Google AdSenseThis is an ad network. (Privacy Policy)
      Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
      Index ExchangeThis is an ad network. (Privacy Policy)
      SovrnThis is an ad network. (Privacy Policy)
      Facebook AdsThis is an ad network. (Privacy Policy)
      Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
      AppNexusThis is an ad network. (Privacy Policy)
      OpenxThis is an ad network. (Privacy Policy)
      Rubicon ProjectThis is an ad network. (Privacy Policy)
      TripleLiftThis is an ad network. (Privacy Policy)
      Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
      Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
      Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
      Statistics
      Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
      ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
      Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)