IDENTITIES Expansion Formulas | (a + b)2 | a2 + 2ab + b2 |
| (a - b)2 | a2 - 2ab + b2 |
| (a+b)(a-b) | a2 - b2 |
| (a+b)2 -(a-b)2 | 4ab |
| (a + b)3 | a3 + 3a2b + 3ab2 + b3 |
| (a - b)3 | a3 - 3a2b + 3ab2 - b3 |
| (a + 1/a )3 | a3 + 1/a3 + 3(a + 1/a) |
| (a - 1/a )3 | a3 - 1/a3 - 3(a - 1/a) |
| (x + a)(x + b) | x2 + ax + bx + ab |
| (x + a)(x - b) | x2 + ax - bx - ab |
| (x - a)(x + b) | x2 - ax + bx - ab |
| (x - a)(x - b) | x2 - ax - bx + ab |
Example 1 : Expand the following :
(i) (4x - 5y)
2 Solve :- By using Identity :-
(a - b)
2 = a
2 - 2ab + b
2 (4x - 5y)
2 = (4x)
2 - 2(4x)(5y) + (5y)
2 =
16x
2 - 40xy + 25y
2 Ans :- 16x
2 - 40xy + 25y
2 (ii) (5a
2 - 4b
3)
2 Solve :- By using Identity :-
(a - b)
2 = a
2 - 2ab + b
2
(5a
2 - 4b
3)
2 = (5a
2)
2 - 2(5a
2)(4b
3) + (4b
3)
2 = 25a
4 - 40a
2b
3 +16b
6 Ans :- 25a
4 - 40a
2b
3 +16b
6 Example 2 :
If a + b = 7 and ab = 12, find the value of (a) a - b (b) a
2 - b
2 Solution :- By using Identity :-
(a + b)
2 - (a - b)
2 = 4ab
= (7)
2 - (a - b)
2 = 4 x 12
49 - 48 = (a - b)
2 (a - b)
2 = +- 1
(a) a- b = +- 1
(b) a
2 - b
2 = (a + b)(a - b) = 7( +- 1) = +- 7