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

*The author is an online writer and math enthusiast who loves to teach readers.*

## Proving a Simple Formula

Why** (a+b) ^{ 2} = a^{2}+b^{2}+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) = a ^{2} +ab + ba + b^{2} = a^{2 }+ 2ab + b^{2}**

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**to calculate the area of the square: we need to calculate the areas of parts 1,2,3,4, and sum up.**fig b,****Calculation**: Please refer to**fig c**.

**Area of part 1 :**

Part 1 is a square of length a.

Therefore the area of part 1 = **a ^{2 }**---------------------------- (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 =** b ^{2 }**----------------------------(iv)

So, Area of square of length (a+b) = (a+b)^{2 }= (i) + (ii) + (iii) + (iv)

Therefore :

(a+b)^{2} = a^{2} + ba + ba +b^{2}

i.e.** (a+b) ^{2 }= a^{2} + 2ab + b^{2}**

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:

Hi guys I'm u r new friend

**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:

9771090194

**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