Identity Verification: a^2−b^2 MCQs Quiz | Class 9

Welcome to the Class IX Mathematics (Code 041) quiz for Unit II: Algebra. This specific exercise focuses on the topic “Identity Verification: a^2−b^2” and covers verifying and applying the difference of squares identity. Test your skills in factorization and expansion using this algebraic rule. Click “Submit Quiz” to check your score and download the complete Answer Key PDF.

Understanding the Difference of Squares Identity

The algebraic identity a^2 – b^2 = (a – b)(a + b) is one of the most fundamental tools in Class 9 Algebra. It states that the difference of the squares of two terms is equal to the product of their sum and their difference. This identity allows for quick factorization of binomials and simplifies mental arithmetic.

Key Concepts

  • Structure: The expression must consist of exactly two terms, separated by a minus sign, where both terms are perfect squares.
  • Verification: To verify, you can multiply the factors: (a-b)(a+b) = a(a+b) – b(a+b) = a^2 + ab – ab – b^2 = a^2 – b^2. The middle terms cancel out.
  • Application: It is used to factorize algebraic expressions like x^2 – 16 and to solve arithmetic problems like 98 x 102 quickly.

Common Examples

Expression (a^2 – b^2) Identified Terms Factors (a – b)(a + b)
x^2 – 25 a = x, b = 5 (x – 5)(x + 5)
49 – y^2 a = 7, b = y (7 – y)(7 + y)
4x^2 – 9y^2 a = 2x, b = 3y (2x – 3y)(2x + 3y)

Quick Revision Notes

  1. Ensure both terms are squares. If you see coefficients like 3x^2 – 12, factor out the common term first: 3(x^2 – 4) = 3(x-2)(x+2).
  2. Do not confuse with (a – b)^2, which expands to a^2 – 2ab + b^2.
  3. Always check if the resulting factors can be factorized further (e.g., x^4 – y^4).

Extra Practice Questions

Try solving these on your own:

  1. Factorize: 16x^2 – 81
  2. Evaluate without direct multiplication: 54^2 – 46^2
  3. Find the value of k if k^2 – 1 = 80 and k > 0.
  4. Expand: (3m + 4n)(3m – 4n)
  5. Factorize completely: x^8 – 1