Why Does (a+b) 2 = a2+b2+2ab?
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Proving a Simple Formula
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) become 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 a length (a+b).
- As per fig b, to calculate the area of the square: we need to calculate the areas 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 the 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 proofs.
Comments
Md akdas on December 25, 2019:
Osm hai
Naetik Tiwari on November 26, 2019:
It's been observed that committing formulae by heart is tentative and so it is of no use later on but working them out practically is of greater use as leaves an indellible mark on one's mind and stays forever.
Okapil on November 13, 2019:
Thank you ever so much
We tend to teach students just "formulas" without demonstrating and what they are need for
Panchi on August 02, 2019:
Thanks
Yashir on July 22, 2019:
Thanks b because of you i can do my project
vikrant singh on March 05, 2019:
good sir but in hindi
Shaik. Faijan on February 20, 2019:
Good one
Ramamani on February 07, 2019:
Can you give
related questions
Thanks on November 01, 2018:
Thanks
Jaswanth on October 14, 2018:
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CHANDRU CHAN on July 31, 2018:
Nice
kathie on July 23, 2018:
thanks
PRADEEP singh on June 15, 2018:
It is actually better for me thanks for it
deepak on June 06, 2018:
good
Bidor Engti on June 06, 2018:
What is the solution of (a+b)2=
Atul kumar pal on May 23, 2018:
Thanks a lot for help
Janani on May 09, 2018:
I am not satisfied with that proof. Any other please!
aparna on April 16, 2018:
nice
shyam on March 31, 2018:
Good
vinay pratap singh on March 10, 2018:
minded
ishika on February 16, 2018:
very nice.........and helpful
Pratima pandey on February 04, 2018:
Answer in hindi
Aisha chowdhary on December 29, 2017:
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
Uttam on December 17, 2017:
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Mosaira Aaliya on November 13, 2017:
Diploma Electrical Engnnering
Mathematics =3 semester Question and Answer send me please VVI Question
Lan on October 24, 2017:
How a2B 2 +aB2
parthasarathi on October 09, 2017:
A+b =a2*bxv
Samay sahu on October 05, 2017:
Tens
Nihal ahmad on September 30, 2017:
Vvvvv good
Ignatius on August 26, 2017:
Could u give me this kind of geometrical explanation for (a+b)³.?
Ritu on August 13, 2017:
Very nice and helpful to me thanx for it..........
mr.krishna rai Gaurishriram (sirjam) on August 07, 2017:
Very Good.
Sam on July 09, 2017:
Nice sir...........""
Rahul raj on July 04, 2017:
How are you friend
Saumya on June 12, 2017:
Thanks for the article . It was very helpful to me
sunil basnet on June 08, 2017:
good
Shiny on May 31, 2017:
Thanks for this but can you give some questions related to this and its answer .please
tutku on May 27, 2017:
hello bawa! today I really found out the mystry of the Universe.
Suraj on April 13, 2017:
Good, like it
pranav Aryan on March 30, 2017:
Very help ful
samyam on March 17, 2017:
thakyou
Likith on January 05, 2017:
Nice explanation it is easy to learn
Hazique on November 27, 2016:
Thank you sir
Sourab on November 02, 2016:
I really liked the proof and I never knew that it can be proved in this way tooooo......
sumati joshi on October 05, 2016:
thanks for this
pasha on August 26, 2016:
How many a and how many b in a+bwhole square..?
kaushik raj on July 27, 2016:
can you plz proo another formula
ASif ali lone on April 22, 2016:
Very useful and interesting
abhishek kumar singh on April 05, 2016:
good job good site
Aarthi on February 08, 2016:
very useful
A on November 02, 2015:
Good job
Angad chauhan on September 05, 2015:
(a-b)2=a2+b2-2ab.
Pravallika on August 19, 2015:
Good
Aryan aman on June 12, 2015:
Spcl thnx
satyam on May 18, 2015:
Tnx useful for me
DHANANJAY KUMAR on April 05, 2015:
THANKS IS VERY HELPFUL
sathya prathap on January 05, 2015:
Thanks. Very useful
yosef on October 16, 2014:
it is very useful for our learnig
varun on October 15, 2014:
How d formula z used in daily life
aanchal on October 11, 2014:
thankyou
sally on October 09, 2014:
tnx was helpful
SACHIN on March 23, 2014:
wwhat awomderfull
SACHIN on March 23, 2014:
wwhat awomderfull
rashmi jain on October 17, 2013:
thanx
zoya khan on October 16, 2013:
Not Bad but it can be shorterand thanks for giving me answer . THANK YOU HUBPAGE.COM.THANK YOU VERY MUCH
Hassan Shahbaz from Islamabad, Pakistan on May 25, 2013:
coooool :D
Loved it :)
(y)
messi jan on May 24, 2013:
very good
najwa on May 24, 2013:
very useful
ashish chandran on February 19, 2013:
explainted very well.
aysha on February 17, 2013:
really helpful proof
Abir on November 29, 2012:
nice prooof
basavaraj on November 22, 2012:
in the same way, do u know (a-b)^2
basavaraj on November 22, 2012:
in the same way, do u know (a-b)^2
Caleb DRC on October 17, 2012:
This is fantastic; I had no idea, and I have never seen it explained in this way. Great job!
Tvrtko on October 13, 2012:
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
urmi on September 29, 2012:
not bad
p l patel on April 04, 2012:
good