The document outlines important algebraic formulas, including identities for the difference and sum of squares and cubes. It provides derivations for several key equations such as (a + b)^2, (a - b)^2, and (a + b + c)^2. The content serves as a concise reference for fundamental algebraic properties and their proofs.

1. Presented by
Abdul Ghaffar
Important Formula
of Algebra

2. 1.a2 – b2 = (a – b)(a + b)
2.(a+b)2 = a2 + 2ab + b2
3.a2 + b2 = (a – b)2 + 2ab
4.(a – b)2 = a2 – 2ab + b2
5.(a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc
6.(a – b – c)2 = a2 + b2 + c2 – 2ab – 2ac + 2bc
7.(a + b)3 = a3 + 3a2b + 3ab2 + b3
8.(a – b) 3 =a3 - 3a2b + 3ab2 - b3
Basis Algeric Formula

3. a2 – b2 = (a – b)(a + b)
Taking Right Hand side
(a – b)(a + b)
a(a – b) + b(a – b)
a2 +ab – ab – b2
a2 – b2
(a+b)2 = a2 + 2ab + b2
Taking left Hand side
(a+b)2
(a + b)(a + b)
a(a + b) + b(a + b)
a2 +ab + ab + b2
a2 + 2ab + b2
a2 + b2 = (a – b)2 + 2ab
Taking Right Hand side
(a – b)2 + 2ab
(a - b)(a - b) +2ab
a(a - b) - b(a - b) +2ab
a2 - ab - ab + b2 +2ab
a2 - 2ab + b2 +2ab
a2 + b2
(a – b)2 = a2 – 2ab + b2
Taking left Hand side
(a – b)2
(a - b)(a - b)
a(a - b) - b(a - b)
a2 - ab - ab + b2
a2 - 2ab + b2
Verify Formula

4. (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc +2ac
Taking left Hand side
(a + b + c)2
(a + b + c) (a + b + c)
a(a + b + c) +b(a + b + c)+c(a + b + c)
a2 + ab + ac + ab + b2 + bc +ac + bc + c2
a2 + 2ab + 2ac + b2 + 2bc + c2
Rearrange
a2+ b2 + c2 + 2ab + 2bc +2ac
Taking left Hand side
(a - b - c)2
(a - b - c) (a - b - c)
a(a - b - c) - b(a - b - c)- c(a - b - c)
a2 - ab - ac - ab + b2 + bc - ac + bc + c2
a2 - 2ab - 2ac + b2 + 2bc + c2
Rearrange
a2+ b2 + c2 - 2ab + 2bc - 2ac
(a - b - c)2 = a2 + b2 + c2 - 2ab + 2bc - 2ac

5. (a + b)3 = a3 + 3a2b + 3ab2 + b3
Taking left Hand side
(a+b)3
(a + b)(a + b) (a + b)
(a+b)2 (a + b)
(a2 + 2ab + b2) (a + b)
a (a2 + 2ab + b2) +b (a2 + 2ab + b2)
a3 + 2a2b + ab2 + a2b+2ab2+ b3
a3 + 3a2b + 3ab2 + b3
or
a3+ b3 +3ab (a+b)
(a - b)3 = a3 - 3a2b + 3ab2 - b3
Taking left Hand side
(a-b)3
(a - b)(a - b) (a - b)
(a- b)2 (a - b)
(a2 - 2ab + b2) (a - b)
a (a2 - 2ab + b2) - b (a2 - 2ab + b2)
a3 - 2a2b + ab2 - a2b +2ab2 - b3
a3 - 3a2b + 3ab2 - b3
or
a3- b3 -3ab (a-b)

6. Thank you