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

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.