Mathematics solution NCERT
Class 9 - Chapter 4: Exploring Algebraic Identities
Q1: Fill in the blanks to complete the following identities.
(i) s² − 11s + 24 = ( ____ )( ____ )
We need two numbers whose product is 24 and sum is -11.
-3 × -8 = 24
-3 + (-8) = -11
Therefore,
s² − 11s + 24 = (s − 3)(s − 8)
Answer: (s − 3)(s − 8)
(ii) ( ____ )(x + 1) = 3x² − 4x − 7
Factor the RHS:
3x² − 4x − 7
= 3x² + 3x − 7x − 7
= 3x(x + 1) − 7(x + 1)
= (3x − 7)(x + 1)
Answer: (3x − 7)
(iii) 10x² − 11x − 6 = (2x − ____)(____ + 2)
Factor the expression:
10x² − 11x − 6
= 10x² + 4x − 15x − 6
= 2x(5x + 2) − 3(5x + 2)
= (2x − 3)(5x + 2)
Answer:
(2x − 3)(5x + 2)
(iv) 6x² + 7x + 2 = ( ____ )( ____ )
6x² + 7x + 2
= 6x² + 3x + 4x + 2
= 3x(2x + 1) + 2(2x + 1)
= (3x + 2)(2x + 1)
Answer:
(3x + 2)(2x + 1)
Q2: Select and use the identity that will help you to find the following products without multiplying directly.
(i) (41)²
= (40 + 1)²
= 40² + 2(40)(1) + 1²
= 1600 + 80 + 1
= 1681
Answer: 1681
(ii) (27)²
= (30 − 3)²
= 900 − 180 + 9
= 729
Answer: 729
(iii) 23 × 17
= (20 + 3)(20 − 3)
= 20² − 3²
= 400 − 9
= 391
Answer: 391
(iv) (135)²
= (100 + 35)²
= 10000 + 7000 + 1225
= 18225
Answer: 18225
(v) (97)²
= (100 − 3)²
= 10000 − 600 + 9
= 9409
Answer: 9409
(vi) 18 × 29
= (20 − 2)(20 + 9)
= 20² + 20(9 − 2) − 18
= 400 + 140 − 18
= 522
Answer: 522
(vii) 34 × 43
= (38.5 − 4.5)(38.5 + 4.5)
= (38.5)² − (4.5)²
= 1482
Answer: 1462
(viii) (205)²
= (200 + 5)²
= 40000 + 2000 + 25
= 42025
Answer: 42025
Q3: Factor the following.
(i) 9a² + b² + 4c² − 6ab + 12ac − 4bc
= (3a)² + (-b)² + (2c)²
+ 2(3a)(-b)
+ 2(3a)(2c)
+ 2(-b)(2c)
= (3a − b + 2c)²
Answer:
(3a − b + 2c)²
(ii) 16s² + 25t² − 40st
= (4s)² + (5t)² − 2(4s)(5t)
= (4s − 5t)²
Answer: (4s − 5t)²
(iii) r² − r − 42
We need two numbers whose product is -42 and sum is -1.
-7 × 6 = -42
-7 + 6 = -1
Therefore,
r² − r − 42
= (r − 7)(r + 6)
Answer: (r − 7)(r + 6)
(iv) 49g² + 14gh + h²
= (7g)² + 2(7g)(h) + h²
= (7g + h)²
Answer: (7g + h)²
(v) 64u² + 121v² + 4w² − 176uv − 32uw + 44vw
= (8u)² + (-11v)² + (-2w)²
+ 2(8u)(-11v)
+ 2(8u)(-2w)
+ 2(-11v)(-2w)
= (8u − 11v − 2w)²
Answer:
(8u − 11v − 2w)²