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Important Algebra Formulas and Identities with Step by Step

By Malay Das June 14, 2021  

Important Algebra Formulas and Identities

Formula - Formula is a mathematical expression or rule.
Identity - An identity is an equation that is true for all the values of the variables.

Formulas

  1. (a + b)2 = a2 + b2 + 2ab
  2. (a - b)2 = a2 + b2 - 2ab
  3. a2 - b2 = (a+b)(a-b)
  4. a2 + b2 = (a+b)2 -2ab
  5. (a+b+c)2 = a 2 + b 2 + c 2 + 2ab + 2bc + 2ac
  6. (a-b-c)2 = a 2 + b 2 + c 2 - 2ab + 2bc - 2ac
  7. (a + b)3 = a3 + b3 + 3a2b + 3ab2
                    = a3 + b3 + 3ab(a+b)
  8. (a - b)3 = a3 - b3 - 3a2b + 3ab2
                    = a3 - b3 - 3ab(a-b)
  9. a3 + b3 = (a+b)(a2 - ab + b2)
  10. a3 - b3 = (a-b)(a2 + ab + b2)

Deducing the Important Formulas

1. a2 = a * a

2. (a + b)2 = (a + b)(a + b)                  [ Back to Formulas ]

                = a2 + ab + ab + b2

                = a2 + 2ab + b2

3. (a – b)2 = (a – b)(a – b)                  [ Back to Formulas ]

               = a2 - ab - ab + b2

               = a2 - 2ab + b2

4 (i). a2 + b2 = (a + b)2 – 2ab                     [ Back to Formulas ]

[As (a + b)2  = a2 + 2ab + b2 and to get a2 + b2  we only need to add – 2ab]

 (ii). a2 + b2 = (a - b)2 + 2ab  

[As (a - b)2  = a2 - 2ab + b2 and to get a2 + b2  we only need to add + 2ab]

5 (i). a2 - b2 = (a - b)2 + 2ab – 2b2                  [ Back to Formulas ]

[As (a - b)2  = a2 - 2ab + b2 and  to get a2 - b2 we only need to add  + 2ab and – 2b2]  = (a - b)2 + 2b (a – b)

          = (a – b) [ (a – b) + 2b]

          = (a – b)(a + b)

 (ii). a2 - b2 = (a + b)2 - 2ab – 2b2

[As (a + b)2  = a2 + 2ab + b2 and  to get a2 - b2 we only need to add  - 2ab and – 2b2]

          = (a + b)2  - 2b (a + b)

          = (a + b) [ (a + b) - 2b]

          = (a + b)(a – b)

6.(a + b + c)2 = (a + b + c)(a + b + c)                  [ Back to Formulas ]

                   = a(a + b + c) + b(a + b + c) + c(a + b + c)

                   = a2 + ab + ac + ab + b2 + bc + ac + bc + c2

                   = a2 + b2 + c2 + ab + ab + ac + ac + bc + bc

                   = a2 + b2 + c2 + 2ab + 2ac + 2bc

7.(a - b - c)2 = (a - b - c)(a - b - c)                  [ Back to Formulas ]

                   = a(a - b - c) - b(a - b - c) - c(a - b - c)

                   = a2 - ab - ac - ab + b2 + bc - ac + bc + c2

                   = a2 + b2 + c2 - ab - ab - ac - ac + bc + bc

                   = a2 + b2 + c2 - 2ab - 2ac + 2bc