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Identities - Part II |
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2. Identity for the square of subtraction of two terms If
a and b are two variables or terms, then the identity for the difference or
subtraction of a and b is (a
- b)2 = a2 -
2ab
+ b2 Let
us try and prove this identity. (a
- b)2 = (a - b) * (a
- b)
= a (a - b) + b(a - b)
= a2 - ab - ba
+ b2 Since
ab = ba
To
verify if this is an identity let
us put a few values of a and b and see if the identity is true. Example 1:
Let a = 3, b = 2. Prove that (a
- b)2 = a2 - 2ab
+ b2 LHS
: ( 3 - 2)2
= (1) 2 = 1 RHS
: 32 - 2 * 3 * 2 + 22
= 9 - 12 + 4 = 13 - 12 = 1 Thus
LHS = RHS The
equation (a - b)2 =
a2 - 2ab
+ b2 is valid for a = 3 and b = 2 Example 2
: Let a = 10, b =0.5 . Prove
that (a - b)2 = a2 - 2ab
+ b2 LHS
: ( 10 ñ 0.5)2 =
(9.5) 2 = 90.25 RHS
: 102 - 2 * 10 * 0.5 + 0.25 = 100 ñ 10 + 0.25 = 90.25 Thus
LHS = RHS The
equation (a - b)2 =
a2 - 2ab
+ b2 is valid for a = 10 and b =0. 5 From
the above three examples, we can see that the identity (a - b)2
= a2 -
2ab + b2
remains valid for any values of a and b including fractions or
decimals numbers. Thus, the square of the subtraction of
two terms = the square of the first term - twice the product of the two
terms + the square of the second term.
Area
of 
KLMN can
be divided as shown in different segments of a and b.
The area -
area
of rectangle OLPS - area
of the square SPMQ = area I - area II - area III - area IV area
I = a * a = a2 area
II = (a ñ b)* b = ab ñ b2 area
III = (a ñ b )* b = ab ñ b2 area
IV = b2 Thus
the area of square KOSR = (a - b)2
= a2 ñ
ab ñ b2 ñ (ab ñ b2
Example
4
: Expand (2x - 3y)2 Using
the identity (a ñ b)2
= a2 ñ 2ab
+ b2 we can write (2x
- 3y)2 = (2x)2
- 2 * 2x * 3y
+(3y)2
= 4x2 - 12xy
+ 9y2 Thus
(2x - 3y)2 = 4x2
- 12xy + 9y2 Example
5
: Find the square of 95. 952
= (100 - 5)2
= 1002 - 2 * 100 * 5+ 52
= 10000 - 1000 + 25
= 9025 Thus
952 = 9025 Example
6
: Find the square of 3/4. (3/4)2
= (1 - º)2
= 12 - 2 * 1
*
1/4 + (1/4)2
= 1 - 1/2 + 1/16
= 1 ñ 0.5 + 0.062 Thus
(3/4)2 = 0.562 In
examples 5 and 6 it is shown how squares
of large numbers can be calculated easily by writing the number as
difference of two easy numbers.
3.
Identity
for addition * subtraction of the two terms
Let
a and b two variables. We
have to find the identity for (a
+ b) *
(a ñ b). Let
us calculate (a + b) * (a ñ b) (a
+ b) *
(a ñ b) = a (a ñ b)
+ b (a ñ b)
= a2 ñ ab + ab -
b2
= a2 ñ b2 Thus
(a + b) *
(a ñ b) = a2 ñ
b2
is the third identity. The
addition of two terms into subtraction of the same two terms is equal to the
square of the first term minus the square of the second term.
Consider
a rectangle USZX. Its one side US is (a
+ b) and the other side is (a ñ b) Area
of rectangle USZX = area of square UVTR - area of the rectangle XWTR
=
a * a ñ a * b + (a ñ b) *
b
= a2
ñ ab + ab ñb2
= a2 ñ b2 Thus
(a + b) * (a ñ b) = a2 ñ b2 Example 1
: Expand ( 2p + q) ( 2p ñq) Using
the identity (a + b)
* (a ñ b)
= a2 -
b2, we can write (
2p + q) ( 2p ñq) = (2p)2
ñ (q) 2
= 4p2 ñ q2 Example 2
: Find 52 * 48 52
= 50 + 2 48
= 50 ñ 2 Using
the identity (a + b) * (a ñ b)
= a2 - b2,
we can find 52 * 48 52
* 48 = (50
+ 2) * (50 ñ 2)
= 502 - 22
= 2500 ñ 4
= 2496 Thus
52 * 48 is 2496. |
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