ML Aggarwal Solutions Class 9 Math Fourth Chapter Factorisation Exercise 4.3

ML Aggarwal Solutions Class 9 Math 4th Chapter Factorisation Exercise 4.3

ML Aggarwal Understanding ICSE Mathematics Class 9 Solutions Fourth Chapter Factorisation Exercise 4.3. APC Solution Class 9 Exercise 4.3.

 

(1) (i) 4x2 – 25y2

Solution:

4x2 – 25y2

= (2x)2 – (5y)2

= (2x + 5y) (2x – 5y)

 

(ii) 9x2 – 1

Solution:

9x2 – 1

= (3x)2 – 12

= (3x + 1) (3x – 1)

 

(2) (i) 150 – 6a2

Solution:

150 – 6a2

= 6 (25 – a2)

= 6 [(5)2 – a2]

= 6 (5+a) (5-a)

 

(ii) 32x2 – 18y2

Solution:

32x2 – 18y2

= (16x2 – 9y2)

= 2 [(4x)2 – (3y)2]

= 2 (4x + 3y) (4x – 3y)

 

(3) (i) (x – y)2 – 9

Solution:

(x – y)2 – 9

= (x – y)2 – 32

= (x – y + 3) (x – y – 3)

 

(ii) 9 (x + y)2 – x2

Solution:

9 (x + y)2 – x2

= [3 (x + y)]2 – x2

= [3 (x + y) – x] [3 (x + y) + x]

= (3x + 3y – x) (3x + 3y + x)

= (2x + 3y) (4x + 3y)

 

(4) (i) 20x2 – 45y2

Solution:

20x2 – 45y2

= 5 (4x2 – 9y2)

= 5 [(2x)2 – (3y)2]

= 5 (2x + 3y) (2x – 3y)

 

(ii) 9x2 – 4 (y + 2x)2

Solution:

9x2 – 4 (y – 2x)2

= (3x)2 – [2 (y – 2x)]2

= (3x + 2y – 4x)

(3x – 2y + 4x)

= (2y – x) (7y – 2y)

 

(5) (i) 2 (x – 2y)2 – 50y2

Solution:

2 (x – 2y)2 – 50y2

= 2 [(x – 2y)2 – 25y2]

= 2 [(x – 2y)2 – (5y)2]

= 2 (x – 2y + 3y) (x – 2y – 5y)

= 2 (x + 3y) (x – 7y)

 

(ii) 32 – 2 (x – 4)2

Solution:

32 – 2 (x – 4)2

= 2 [16 – (x – 4)2]

= 2 [42 – (x – 4)2]

= 2 (4 + x – 4) (4 – x + 4)

= 2x (8 – x)

 

(6) (i) 108a2 – 3 (b – c)2

Solution:

108a2 – 3 (b – c)2

= 3 [36a2 – (b – c)2]

= 3 [(6a)2 – (b – c)2]

= 3 (6a – b + c) (6a + b + c)

 

(ii) πa5 – π3ab2

Solution:

πa5 – π3ab2

= πa (a4 – π2b2)

= πa [(a2)2 – (πb)2]

= πa (a2 – πb) (a2 + πb)

 

(7) (i) 50x2 – 2 (x – 2)2

Solution:

50x2 – 2 (x – 2)2

= 2 [25x2 – (x – 2)2]

= 2 [(5x)2 – (x – 2)2]

= 2 (5x + x – 2) (5x – x + 2)

= 2 (6x – 2) (4x + 2)

= 2 × 2 × 2 (3x – 1) (2x + 1)

= 8 (3x – 1) (2x + 1)

 

(ii) (x – 2) (x + 2) + 3

Solution:

(x – 2) (x + 2) + 3

= x2 – 4 + 3

= x2 – 1

= x2 – 12

= (x + 1) (x – 1)

 

(8) (i) x – 2y – x2 + 4y2

Solution:

x – 2y – x2 + 4y2

= x – 2y – 1 (x2 – 4y2)

= x – 2y – 1 [x2 – (2y)2]

= x – 2y – 1 (x – 2y) (x + 2y)

= (x – 2y) (1 – x – 2y)

 

(ii) 4a2 – b2 + 2a + b

Solution:

4a2 – b2 + 2a + b

= (2a)2 – b2 + 2a + b

= (2a + b) (2a – b) + 2a + b

= (2a + b) (2a – b + 1)

 

(9) (i) a (a – 2) – b (b – 2)

Solution:

a (a – 2) – b (b – 2)

= a2 – 2a – b2 – 2b

= a2 – b2 – 2a – 2b

= (a + b) (a – b) – 2 (a + b)

= (a + b) (a – b – 2)

 

(ii) a (a – 1) – b (b – 1)

Solution:

a (a – 1) – b (b – 1)

= a2 – a – b2 – b

= a2 – b2 – a – b

= (a + b) (a – b) – 1 (a + b)

= (a + b) (a – b – 1)

 

(10) (i) 9 – x2 + 2xy – y2

Solution:

9 – x2 + 2xy – y2

= 9 – 1 (x2 – 2xy + y2)

= 9 – 1 (x – y)2

= (3) – (x – y)2

= (3 + x – y) (3 – x + y)

 

(ii) 9x4 – (x2 + 2x + 1)

Solution:

= (3x2)2 – (x + 1)2

= (3x2 + x + 1) (3x2 – x – 1)

 

(11) (i) 9x4 – x2 – 12x – 36

Solution:

9x4 – x2 – 12x – 36

= 9x4 – 1 (x2 + 12x + 36)

= 9x4 – 1 (x2 + 2 × 6 × x + 62)

= 9x4 – 1 (x + 6)2

= 9x4 – (x + 6)2

= (3x2)2 – (x + 6)2

= (3x2 + x + 6) (3x2 – x – 6)

 

(ii) x3 – 5x2 – x + 5

Solution:

x3 – 5x2 – x + 5

= x2 (x – 5) – 1 (x – 5)

= (x – 5) (x2 – 1)

= (x – 5) (x + 1) (x – 1)

 

(12) a4– b4 + 2b– 1

Solution:

a4 – b4 + 2b2 – 1

= a4 – 1 (b4 – 2b2 + 1)

= a4 – 1 [(b2)2 – 2 × b2 × 1 + 12]

= a4 – 1 (b2 – 1)2

= a4 – (b2 – 1)2

= (a2)2 – (b2 – 1)2

= (a2 – b2 + 1) (a2 + b2 – 1)

= [(a + b) (a – b) + 1] [a2 + (b + 1) (b – 1)]

 

(ii) x3 – 25x

Solution:

x3 – 25x

= x (x2 – 25)

= x [x2 – 52]

= x (x + 5) (x – 5)

 

(13) (i) 2x4 – 32

Solution:

2x4 – 32

= 2 (x4 – 16)

= 2 [(x2)2 – 42]

= 2 (x2 + 4) (x2 – 4)

= 2 (x2 + 4) [x – 22]

= 2 (x2 + 4) (x + 2) (x – 2)

 

(ii) a2 (b + c) – (b + c)3

Solution:

a2 (b + c) – (b + c)3

= (b + c) [a2 – (b + c)2]

= (b + c) (a + b + c) (a – b – c)

 

(14) (i) (a + b)3 – a – b

Solution:

(a + b)3 – a – b

= (a + b)3 – 1 (a + b)

= (a + b) [(a + b)2 – 1]

= (a + b) (a + b + 1) (a + b – 1)

 

(ii) x2 – 2xy + y2 – a2 – 2ab – b2

Solution:

x2 – 2xy + y2 – a2 – 2ab – b2

= (x – y)2 – 1 (a2 + 2ab + b)2

= (x – y)2 – (a + b)2

= (x – y + a + b) (x – y – a – b)

 

(15) (i) (a2 – b2) (c2 – d2) – 4abcd

Solution:

(a2 – b2) (c2 – d2) – 4abcd

= a2c2 – a2d2 – b2c2 + b2d2 – 4abcd

= a2c2 – 4abcd + b3d3 – a2d3 – b2c2

= a2c2 – 2abcd + b2 d2 – a3d2 – 2abcd – b3c2

= a3c3 – 2abcd + b2d2 – 1 (a2d2 + 2abcd + b2c2)

= (ac)2 – 2ac.bd + (bd)2 -1 [(ad)2 + 2ad.bc + (bc)2]

= (ac – bd)2– (ad + bc)2

= (ac – bd + ad + bc) (ac – bd – ad – bc)

 

(15) (ii) 4x2 – y2 – 3xy + 2x – 2y

Solution:

4x2 – y2 – 3xy + 2x – 2y

= x2 – y2 + 3x2 – 3xy + 2x – 2y

= (x + y) (x – y) + 3x (x – y) + 2 (x – y)

= (x – y) (x + y + 3x + 2)

= (x – y) (4x + y + 2)

 

(16) (i) x2 + 1/x2 – 11

Solution: 

x2 + 1/x2 – 11

= x2 + 1/x2 – 2 – 9

= (x – 1/x)2– 9

= (x – 1/x)2 – 32

= (x – 1/x + 3) (x – 1/x – 3)

 

(ii) x4 + 5x2 + 9

Solution:

x4 + 5x2 – 9

= x4 + 6x2 + 9 – x2

= (x2)2 + 2.x2× 3 + 32 – x2

= (x2 + 3)2 – x2

= (x2 + 3 + x) (x2 + 3 – x)

 

(17) (i) a4 + b4 – 7a2b2

Solution: 

a4 + 2a2b2 + b2 – 9a2b2

= (a2 + b2)2 – (3ab)2

= (a2 + b2 + 3ab) (a2 + b2 – 3ab) – Ans.

= (a2 + b2 + 2ab + ab) (a2 + b2 – 2ab – ab)

= [(a + b)2 + ab] [(a – b)2 – ab]

= ab [(a + b)2 + 1] [(a – b)2 – 1]

= ab [(a + b)2 + 1] [(a + b) (a – b) – 1]

 

(ii) x4 – 14x2 + 1

Solution:

x4– 14x2 + 1

= x4 + 2x2 + 1 – 16x2

= (x2)2 + 2.x2 + 12 – (4x)2

= (x2 + 1)2 – (4x)2

= (x2 + 1 – 4x) (x2 + 1 + 4x)

 

(18) Express each of the following as the difference of two squares:

(i) (x2 – 5x + 7) (x2 + 5x + 7)

Solution:

(x2 – 5x + 7) (x2 + 5x + 7)

= (x2 + 7 – 5x) (x2 + 7 + 5x)

= (x2 + 7)2 – (5x)2

 

(ii) (x2 – 5x + 7) (x2 – 5x – 7)

Solution:

(x2 – 5x + 7) (x2 – 5x – 7)

= (x2 – 5x)2 – 72

 

(iii) (x2+ 5x – 7) (x2 – 5x + 7)

Solution:

(x2+ 5x – 7) (x2 – 5x + 7)

= [(x2 + 1 (5x – 7)] [x2 – 1) (5x – 7)]

= [x2 + (5x – 7)] [x2 – (5x – 7)]

= (x2)2 – (5x – 7)2

 

(19) Evaluate the following by using factors:

(i) (979)2 – (21)2

Solution:

(979)2 – (21)2

= (979 + 21) (979 – 21)

= 1000 × 958

= 958000

 

(ii) (99.9)2 – (0.1)2

Solution:

(99.9)2 – (0.1)2

= (99.9 + 1) (99.9 – 0.1)

= 100 × 99.8

= 9980

Updated: June 17, 2022 — 2:39 pm

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  1. Yes,very best answers.

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