Class 9 Mathematics

Chapter 4 Solutions — Exploring Algebraic Identities

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Solutions

Overview

Step-by-step NCERT solutions for Exploring Algebraic Identities (Chapter 4, NCERT Class 9 Mathematics) — the full working for every question, not just the final answer. You can also read the Exploring Algebraic Identities textbook chapter.

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What these solutions cover

All 25 questions in Exploring Algebraic Identities are solved in the PDF. Here's what's inside, exercise by exercise:

Visualising Identities — using (a + b)² = a² + 2ab + b²

1.
Using the identity (a + b)² = a² + 2ab + b², expand the following:
- (i) (7x + 4y)²
- (ii) (7x/5 + 3y/2)²
- (iii) (2.5p + 1.5q)²
- (iv) (3s/4 + 8t)²
- (v) (x + 1/(2y))²
- (vi) (1/x + 1/y)²
2.
Using the same identity (a + b)² = a² + 2ab + b², find the values of:
- (i) (64)²
- (ii) (105)²
- (iii) (205)²

Factorisation Using (a + b)² and (a − b)² Identities

1.
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
2.
Find the values of the following using the identity (a − b)² = a² − 2ab + b²:
- (i) (79)²
- (ii) (193)²
- (iii) (299)²

More Identities — (a + b + c)² and (a − b)²

1.
Find the following squares using one of the above identities. Determine which identity makes each calculation easier.
- (i) 117²
- (ii) 78²
- (iii) 198²
- (iv) 214²
- (v) 1104²
- (vi) 1120²
2.
Factor using suitable identities:
- (i) 16y² − 24y + 9
- (ii) (9/4)s² + 6st + 4t²
- (iii) m²/9 + mk/3 + k²/4 + 3nk + 2mn + 9n²
- (iv) p²/16 − 2 + 16/p²
- (v) 9a² + 4b² + c² − 12ab + 6ac − 4bc
3.
Expand the following using the identity (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca:
- (i) (p + 3q + 7r)²
- (ii) (3x − 2y + 4z)²
Is this an identity? (a + b − c)² + (a − b + c)² + (a − b − c)² = 2a² + 2b² + 2c²

Factorisation Using (x + a)(x + b) = x² + (a + b)x + ab and Related Identities

1.
Fill in the blanks to complete the following identities:
- (i) s² − 11s + 24 = (___)(___)
- (ii) (___)( x + 1) = 3x² − 4x − 7
- (iii) 10x² − 11x − 6 = (2x − ___)(___ + 2)
- (iv) 6x² + 7x + 2 = (___)(___)
2.
Select and use the identity that will help you to find the following products without multiplying directly:
- (i) (41)²
- (ii) (27)²
- (iii) 23 × 17
- (iv) (135)²
- (v) (97)²
- (vi) 18 × 29
- (vii) 34 × 43
- (viii) (205)²
3.
Factor the following:
- (i) 9a² + b² + 4c² − 6ab + 12ac − 4bc
- (ii) 16s² + 25t² − 40st
- (iii) r² − r − 42
- (iv) 49g² + 14gh + h²
- (v) 64u² + 121v² + 4w² − 176uv − 32uw + 44vw

Simplification of Rational Expressions Using Algebraic Identities

1.
Simplify the following rational expressions assuming that the expressions in the denominators are not equal to zero:
- (i) (3p² − 3pq − 18q²) / (p² + 3pq − 10q²)
- (ii) (n³ − 3n²m + 3nm² − m³) / (5m² − 10mn + 5n²)
- (iii) (w³ − v³ + x³ + 3wvx) / (w² + v² + x² − 2wv − 2vx + 2wx)
- (iv) (4y² − 20yz + 25z²) / (25z² − 4y²)
- (v) (x² + x − 6)(x² − 7x + 12) / ((x² − 6x + 8)(x² − 9))
- (vi) (p⁴ − 16) / (p² − 4p + 4)

Exercises

1.
Use suitable identities to find the following products:
- (i) (–3x + 4)²
- (ii) (2s + 7)(2s – 7)
- (iii) (p² + 1/2)(p² – 1/2)
- (iv) (2n + 7)(2n – 7)
- (v) (s – 2t)(s² + 2st + 4t²)
- (vi) (1/(2r) – 4r)²
- (vii) (–3m + 4k – l)²
- (viii) (x – y/3)³
- (ix) ((7/2)k – (2/3)m)³
2.
Find the values using suitable identities:
- (i) 17 × 21
- (ii) 104 × 96
- (iii) 24 × 16
- (iv) 147³
- (v) 199³
- (vi) 127³
- (vii) (–107)³
- (viii) (–299)³
3.
Factor the following algebraic expressions:
- (i) 4y² + 1 + 1/(16y²)
- (ii) 9m² – 1/(25n²)
- (iii) 27b³ – 1/(64b³)
- (iv) x² + 5x/6 + 1/6
- (v) 27u³ – 1/125 – 27u²/5 + 9u/25
- (vi) 64y³ + z³/125
- (vii) p³ + 27q³ + r³ – 9pqr
- (viii) 9m² – 12m + 4
- (ix) 9x³ – (8/3)y³ + z³/3 + 6xyz
- (x) 4x² + 9y² + 36z² + 12xz + 36yz + 24xy
- (xi) 27u³ – 1/216 – 9u²/2 + u/4
4.
Simplify the following (assume denominators are not equal to 0):
- (i) (4x² + 4x + 1)/(4x² – 1)
- (ii) 9(3a³ – 24b³)/(9a² – 36b²)
- (iii) (s³ + 125t³)/(s² – 2st – 35t²)
5.
Find possible expressions for the length and breadth of each of the following rectangles whose areas are given by the following expressions in square units:
- (i) 25a² – 30ab + 9b²
- (ii) 36s² – 49t²
6.
Find possible expressions for the length, breadth, and height of each of the following cuboids whose volumes are given by the following expressions in cubic units:
- (i) 6a² – 24b²
- (ii) 3ps² – 15ps + 12p
The village playground is shaped as a square of side 40 metres. A path of width s metres is created around the playground for people to walk. Find an expression for the area of the path in terms of s.
If a number plus its reciprocal equals 10/3, find the number.
A rectangular pool has area 2x² + 7x + 3 square hastas. If its width is 2x + 1 hastas, find its length. (Hasta was a unit used to measure length.)
If both x – 2 and x – 1/2 are factors of px² + 5x + r, show that p = r.
If a + b + c = 5 and ab + bc + ca = 10, then prove that a³ + b³ + c³ – 3abc = –25.
By factoring the expression, check that n³ – n is always divisible by 6 for all natural numbers n. Give reasons.
13.
Find the value of:
- (i) x³ + y³ – 12xy + 64, when x + y = –4
- (ii) x³ – 8y³ – 36xy – 216, when x = 2y + 6

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