Let us try to find the square of the number 102. The square of a number, as we know, is the product of the number with itself. One way to do this is by writing the numbers one below the other, and then multiplying them as we normally do. The other way is to break the numbers and then apply distributive property. This will make our work much easier.
Let us see how.
1022 = 102 × 102
= (100 + 2) (100 + 2)
= 100 (100 + 2) + 2 (100 + 2)
= 100 × 100 + 100 × 2 + 2 × 100 + 2 × 2
= 10000 + 200 + 200 + 4
Observing the similar expressions as above, we obtain the following identities.
(a + b)2 = a2 + 2ab + b2
(a − b)2 = a2 − 2ab + b2
To understand the proof of these identities, look at the following video.
Deriving the identities geometrically:
These identities can be derived by geometrical construction as well. Let us learn the same.
(1) (a + b)2 = a2 + 2ab + b2:
Let us consider a square ABCD whose each side measures (a + b) unit.
It can be seen that, we have drawn two line segments at a distance of a unit from A such that one is parallel to AB and other is parallel to AD.
Also, the figure is divided into four regions named as I, II, III and IV.
∴ Area of square ABCD = (a + b)2 sq. unit ...(i)
Region I is a square of side measuring a unit.
∴ Area of region I = a2 sq. unit ...(ii)
Each of regions II and III is a rectangle having length and breadth as a unit and b unit respectively.
∴ Area of region II = ab sq. unit ...(iii)
Area of region III = ab sq. unit ...(iv)
Region IV is a square of side measuring b unit.
∴ Area of region IV = b2 sq. unit ...(v)
From the figure, we have
Area of square ABCD = Area of region I + Area of region II + Area of region III + Area of region IV
On substituting the values from (i), (ii), (iii), (iv) and (v), we get
(a + b)2 = a2 + ab + ab + b2
|⇒ (a + b)2 = a2 + 2ab + b2|
(2) (a − b)2 = a2 − 2ab + b2:
Let us consider a square EFGH whose each side measures a unit.
It can be seen that,...
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