ALGEBRAIC  IDENTITIES 

                                                 

 

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

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

              (a + b)(a – b)  = a2 – b2 

              (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

              (x + a)(x + b) = x2 + (a + b) x + ab

              (a + b)3 = a3 + b3 + 3ab (a + b)

              (a – b)3 = a3 – b3 – 3ab (a – b)

              a3 + b3 + c3– 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca)

                                                                                                                                

 Definition Of Algebraic identities

Algebra is the branch of mathematics that helps in the representation of problems or 

situations in the form of mathematical expressions. It involves variables like x, y, z, and 

mathematical operations like addition, subtraction, multiplication, and division to form a 

meaningful mathematical expression

An algebraic identity is an equality that holds for any values of its variables.

For example, the identity$(�+�{)}^2={�}^2+2��+{�}^2$
  (a + b)2 = a2 + 2ab + b2holds for all values of $�$a and $�$b.

Since an identity holds for all values of its variables, it is possible to substitute instances of one side 

of the equality with the other side of the equality. For example, because of the identity above, we can 

replace any instance of $(�+�{)}^2$ with  a2 + 2ab + b2  and vice versa

${�}^2+2��+{�}^{\mathrm{2<span\; face="Soleil,\; Arial,\; sans-serif"\; style="font-size:\; 15px;">and\; vice\; versa.</span>}}$

Clever use of identities offers shortcuts to many problems by making the algebra easier to 

manipulate. Below are lists of some common algebraic identities

Example of using Identity 1 ;   (a + b)2 = a2 + 2ab + b2

1.   calculate the square the 32

   

     Answer

 

              32 =( 30 +2)

              (30+2)2= (30)2 + 2(30)(2) + (2)2

             (30+2)2= 900 + 1200 + 4

             (30+2)2= 2104

          Hence, square of 54 is 2104

Example of using  Identity 2 ;  (a – b)2 = a2 – 2ab + b2

2.    Calculate the square the 73

               Answer.

            73 = (80 - 7)

            Here, a=70 and b=3

           (73)2= (80 - 7)2

                  Applying the equation,

           (80 - 7)2= (80)2 - 2(80)(7) + (7)2

            (80 - 7)2= 6400 - 1120 + 49

            (80 - 7)2= 5329

            Hence, square of 73 is 5329.

Example of using Identity 3 ;  (a + b)(a – b)  = a2 – b2 

3. Calculate 47 × 53

       Answer

          47 = (50 - 3) whereas 53 = (50 + 3). So,

          47 × 53 = (50 – 3) (50 + 3)

         Here, a = 50 and b = 3

         Applying the identity

         47 × 53 = (50)2 – (3)2

        47 × 53 = 2500 – 9

        47 × 53 = 2491

                                                                                                                                             

  

                HOW TO DO PROBLEMS USING ALGEBRAIC 

   IDENTITIES