Learn Algebra Formula 

Learn Algebra Formula

Learn How algebra formulas are derived

1.(a-b) ² = (a-b) (a-b)

         = a. a- a. b- b. a+ b .b

        = a²-ab-ba+b²

           = a²-2ab+b²

2.(a + b) ^{2 =} (a+ b) (a+ b)

             = a. a+ a. b + b. a+ b .b

             = a²+ab+ba+b²

             = a²+2ab+b²

3.     3.(a+ b)²-2ab = a²+b²

 

      {(a+ b) (a+ b)}-2ab                       ———left side of equal symbol

     {( a. a+ a. b+ b .a + b .b)}-2ab

     (a²+ab+ba+b²)-2ab

    (a²+2ab+ b²)-2ab

   a²+2ab+ b²-2ab                                 —–open the bracket

      a²+b²        

3.      4.   (a-b)(a +b) = a²– b²

(a- b)(a +b)                              ———left side of equal symbol          

a.     a+ a. b -b. a-b. b

a²+ab-ab-b²                           —- +ab and –ab cancelled

a²– b²

3.      5. (a+ b+ c)² = a²+b²+c²+2ab+2bc+2ac

     ( a+ b+ c) ( a+ b+ c)                             —- left side of equal symbol

    (a .a +a .b +a .c +b. a+ b .b+ b .c+ c .a +c .b  +c .c)

    a²+ab+ac+ba+b²+ bc+ca+cb+c²

     a²+b²+c²+2ab+2bc+2ca                    — ab= ba ,  bc = cb and ca =ac

3.    6.(a-b-c)²=a²+b²+c²-2ab+2bc-2ac

              (a-b-c)² = (a-b-c)(a-b-c)                    —we can write like this.

              =(a. a- a. b-a. c-b. a+ b. b +b .c – c .a+ c .b + c .c

             =a²-ab-ac-ba+b²+bc-ca+cb+c²

             =a²+b²+c²-2ab+2bc-2ac         —our formula

7.    (a+ b)³= (a+ b)(a+ b)(a+ b)

                  = {(a+ b) (a+ b)} (a+ b) 

                  = (a. a+ a. b+ b. a+ b. b) (a+ b)

                  = (a^{2 +} ab+ba+b²⁾ (a + b)

                   = (a²+2ab+b2) (a +b)

                   = a².a+a².b+2ab.a+2ab.b+b².a+b².b

                   = a³+a²b+2a²b+2ab²+b²a+b³

                  = a³+3a²b+3ab²+b³        ———our formula

following this we can derive all the formulas of algebra.

Algebraic Expressions:- 

 1. 2x means x + x, i.e. double of x. Broadly, if m is a positive whole number, then mx      means   m times the x. mx is also called the product of m and x.

In x m, m is called exponent and x is called base. Later, the meanings of mx and xm are expanded and are also given in those situations when m is any number of minus, different, irrational etc.

Laws of Exponents: 

1.(x)^m .(x)ⁿ =(x)^{m+ n}

2. (x)^m/(x)ⁿ =(x)^{m- n}

^3. (x y)^m= x ^m y ^m

^4. (x ^m)ⁿ = x ^{m n}

Basic rules and properties of Algebra. 

 

(1) Closure: If a set has two elements x and y, then x * y is also an element of the same set.

(2) Commutativity:  x * y = y * x

(3) Associativity: If x, y, z are components of a set, then (x * y) * z = x * (y * z)

4) Existence of identity: The set must have such a component that a * e = e * a = a

(5) Existence of inverse: In a set, any element corresponding to any component a has a-1 such that a * a-1 = a-1 * a = e

(6) Distribution rules against first operation and second operation: x (y * z) = (x y) * (x z) and (y* z) x = (y x) * (z x)