Algebraic Expressions


An algebraic expression is an expression which consists of variables and constants. In expressions, a variable can take any value while a constant has its fixed value.

Algebraic Identities


An identity is an equality which is true for all values of variables.




Some Identities



  • (a + b)2 = a2 + 2ab + b2
  • (a – b)2 = a2 – 2ab + b2
  • a2 – b2 = (a + b)(a – b)
  • (x + a)(x + b) = x2 + (a + b) x + ab
  • (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
  • (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)




Evaluate each of the following using identities:

(i) (2x − 1/x)2
(ii) (2x + y)(2x - y)
(iii) (a2b − ab2)2
(iv) (a - 0.1)(a + 0.1)
(v) (1.5x2 − 0.3y2)(1.5x2 + 0.3y2)

Solution:

(i) Given,
(2x − 1/x)2 = (2x)2 + (1/x)2 − 2 × 2x × 1/x
(2x − 1/x)2 = 4x2 + 1/x2 – 4      [∴  (a − b)2 = a2 + b2 − 2ab]
Where, a = 2x, b = 1/x
∴  (2x − 1/x)2 = 4x2 + 1/x2 − 4

(ii) Given,
(2x + y)(2x - y)
= (2x)2 − (y)2      [ ∴ (a + b)(a − b) = a2 − b2]
= 4x2 − y2
∴  (2x + y)(2x − y) = 4x2 − y2

 (iii) Given,
(a2b − ab2)2
= (a2b)2 + (ab2)2 − 2 × a2b × ab2            [∴ (a − b)2 = a2 + b2 − 2ab]
Where, a = a2b, b = ab2
= a4b2 + b4a2 − 2a3b3
(a2b − ab2)2 = a4b2 + b4a2 − 2a3b3

 (iv)  Given,
(a - 0.1)(a + 0.1)
= a2 − (0.1)2                  [∴ (a + b)(a − b) = a2 − b2]
Where, a = a and b = 0.1
= a2 − 0.01
∴ (a − 0.1)(a + 0.1) = a2 − 0.01

(v)  Given,
(1.5x2 − 0.3y2)(1.5x2 + 0.3y2)
= (1.5x2)2 − (0.3y2)2      [∴ (a + b)(a − b) = a2 − b2]
Where, a = 1.5x2, b = 0.3y2
= 2.25x4 − 0.09y4
∴ (1.5x2 − 0.3y2)(1.5x2 + 0.3y2) = 2.25x4 − 0.09y4

Algebraic Identities - Class 9th - RD Sharma Exercises with Solution








Algebraic Identities - Class 9th - Sample Question Paper





Solutions