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NCERT Class 9 Mathematics Exercise Set 4.2 Solutions: Exploring Algebraic Identities

AI-written, checked on 23 Oct 2026

Solutions written by Shishya, checked — not the board's official answers

Step-by-step working for every question of Exercise Set 4.2 in NCERT's Class 9 Mathematics book, chapter “Exploring Algebraic Identities”, written by Shishya. Each question is in our own short words; open the NCERT chapter for the exact wording and figures.

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  1. Ex 4.2 Q1

    Factor completely: (i) 9x² + 24xy + 16y², (ii) 4s² + 20st + 25t², (iii) 49x² + 28xy + 4y², (iv) 64p² + (32/3)pq + (4/9)q², (v) 3a² + 4ab + (4/3)b², (vi) (9/5)s² + 6sv + 5v² (for (v) and (vi), try taking out a common factor first).

    1. (i) 9x² + 24xy + 16y² = (3x)² + 2(3x)(4y) + (4y)²
    2. (i) Using (a + b)² = a² + 2ab + b², the expression is (3x + 4y)²
      why?

      Identities such as (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b² and a² − b² = (a + b)(a − b) are true for every value of the letters, so they can replace long multiplication.

    3. (ii) 4s² + 20st + 25t² = (2s)² + 2(2s)(5t) + (5t)²
    4. (ii) Using (a + b)² = a² + 2ab + b², the expression is (2s + 5t)²
      why?

      Identities such as (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b² and a² − b² = (a + b)(a − b) are true for every value of the letters, so they can replace long multiplication.

    5. (iii) 49x² + 28xy + 4y² = (7x)² + 2(7x)(2y) + (2y)²
    6. (iii) Using (a + b)² = a² + 2ab + b², the expression is (7x + 2y)²
      why?

      Identities such as (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b² and a² − b² = (a + b)(a − b) are true for every value of the letters, so they can replace long multiplication.

    7. (iv) 64p² + (32/3)pq + (4/9)q² = (8p)² + 2(8p)((2/3)q) + ((2/3)q)²
    8. (iv) Using (a + b)² = a² + 2ab + b², the expression is (8p + (2/3)q)²
      why?

      Identities such as (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b² and a² − b² = (a + b)(a − b) are true for every value of the letters, so they can replace long multiplication.

    9. (v) 3a² + 4ab + (4/3)b² = 3(a² + (4/3)ab + (4/9)b²)
    10. (v) Inside the bracket, a² + 2(a)((2/3)b) + ((2/3)b)² is a perfect square
    11. (v) The expression is 3(a + (2/3)b)²
    12. (vi) (9/5)s² + 6sv + 5v² = (1/5)(9s² + 30sv + 25v²)
    13. (vi) Inside the bracket, (3s)² + 2(3s)(5v) + (5v)² is a perfect square
    14. (vi) The expression is (1/5)(3s + 5v)²

    Answer: (i) (3x + 4y)²; (ii) (2s + 5t)²; (iii) (7x + 2y)²; (iv) (8p + (2/3)q)²; (v) 3(a + (2/3)b)²; (vi) (1/5)(3s + 5v)²

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  2. Ex 4.2 Q2

    Use (a − b)² = a² − 2ab + b² to find (i) 79², (ii) 193², (iii) 299².

    1. (i) Write 79 as (80 − 1) and apply the identity (a − b)² = a² − 2ab + b².
      why?

      Identities such as (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b² and a² − b² = (a + b)(a − b) are true for every value of the letters, so they can replace long multiplication.

    2. (i) Calculate 80² − 2(80)(1) + 1² = 6400 − 160 + 1.
    3. (i) Simplify the expression to get 6241.
      why?

      Simplifying (combining like terms, cancelling common factors) changes how the expression looks, not its value, so the next step is easier.

    4. (ii) Write 193 as (200 − 7) and apply the identity (a − b)² = a² − 2ab + b².
      why?

      Identities such as (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b² and a² − b² = (a + b)(a − b) are true for every value of the letters, so they can replace long multiplication.

    5. (ii) Calculate 200² − 2(200)(7) + 7² = 40000 − 2800 + 49.
    6. (ii) Simplify the expression to get 37249.
      why?

      Simplifying (combining like terms, cancelling common factors) changes how the expression looks, not its value, so the next step is easier.

    7. (iii) Write 299 as (300 − 1) and apply the identity (a − b)² = a² − 2ab + b².
      why?

      Identities such as (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b² and a² − b² = (a + b)(a − b) are true for every value of the letters, so they can replace long multiplication.

    8. (iii) Calculate 300² − 2(300)(1) + 1² = 90000 − 600 + 1.
    9. (iii) Simplify the expression to get 89401.
      why?

      Simplifying (combining like terms, cancelling common factors) changes how the expression looks, not its value, so the next step is easier.

    Answer: (i) 6241; (ii) 37249; (iii) 89401

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