Class 8 RD Sharma Solutions- Chapter 7 Factorization – Exercise 7.6

Question 1) 4x² + 12xy + 9y²

Solution:

(2x)² + 2 × 2x × 3y + (3y)²

⇒ (2x + 3y)² [(a + b)² = a² + b² + 2ab]

⇒ (2x + 3y) (2x + 3y)

Question 2) 9a² – 24ab + 16b²

Solution:

(3a)² – 2 × 3a × 4b + (4b)²

⇒ (3a – 4b)² [(a + b)² = a² + b² + 2ab]

⇒ (3a – 4b) (3a – 4b)

Question 3) p²q² – 6pqr + 9r² 

Solution:

(pq)² – 2 x pq x 3r + (3r)²

⇒ (pq – 3r)² [(a + b)² = a² + b² + 2ab]

⇒ (pq -3r) (pq -3r)    

Question 4) 36a² + 36a + 9

Solution:

9(4a² + 4a + 1)

⇒ 9{(2a)² + 2 × 2a × 1 + (1)²} 

⇒ 9(2a + 1)²                                  [(a + b)² = a² + b² + 2ab]

⇒ 9 (2a + 1)(2a + 1)

Question 5) a² + 2ab + b² – 16

Solution:

a² + 2ab + b² – 16

⇒ (a + b)² – (4)²

⇒ (a + b + 4) (a + b – 4)

Question 6) 9z² – x² + 4xy – 4y²

Solution:

9z² – x² + 4xy – 4y²

⇒ 9z² – (x² – 4xy + 4y²)

⇒ 9z² – (x² – 2x × 2y + (2y)²)                  [(a + b)² = a² + b² + 2ab]

⇒ 9z² – (x – 2y)²

⇒ (3z)² – (x – 2y)²

⇒ (3z – (x – 2y)) (3z + (x – 2y))

⇒ (3z – x + 2y) (3z + x – 2y)

Question 7) 9a⁴ – 24 a²b² + 16b⁴ – 256

Solution:

9a⁴ – 24 a²b² + 16b⁴ – 256

⇒ (9a⁴ – 24a²b² + 16b⁴ ) – 256

⇒ [(3a²)² – 2 × 3a² × 4b²] – 256                          

⇒ (3a² – 4b²)² – (16)²

⇒ (3a² – 4b² – 16)(3a² – 4b² + 16)

Question 8) 16 – a⁶ + 4a³b³ – 4b⁶

Solution:

16 – (a⁶ – 4a³b³ + 4b⁶)

⇒ 16 – [(a³)² – 2 × a³ × 2b³ + (2b³)²]

⇒ 16 – (a³– 2b³)²

⇒ (4)² – (a³ – 2b³)²

⇒ (4 – a³ + 2b³)(4 + a³ – 2b³)

Question 9) a² – 2ab + b² – c²

Solution:

(a² – 2ab + b²) – c²

⇒ (a² – 2 × a × b + b²) – c²

⇒ (a – b)² – c²

⇒ (a – b – c) (a – b + c)

Question 10) X² + 2X + 1 – 9Y²

Solution:

X² + 2X + 1 – 9Y²

⇒ (X² + 2 × X × 1 + 1² ) – 9Y²

⇒ (X + 1)² – (3Y)²

⇒ (X + 1 – 3Y) (X + 1 + 3Y)

Question 11) a² + 4ab + 3b²

Solution:

a² + 4ab + 4b² – b²

⇒ (a²+ 2 × 2b × a + (2b)²) – b²

⇒ (a + 2b)² – b²

⇒ (a + 2b – b) (a + 2b+ b)

⇒ (a + b) (a + 3b)

Question 12) 96 – 4x – x²

Solution:

100-4 -4x -x²

⇒ 100 – (x² + 4x + 4)

⇒ (10)² – (x + 2)²

⇒ (10 – x – 2) (10 + x + 2)

⇒ (8 – x) (12 + x)

Question 13) a⁴ + 3a² + 4

Solution: 

a⁴ + 4a² – a² + 4

⇒ [(a²)² + 2 × 2 × a² + (2)²] – a²

⇒ (a² + 2)² – a²

⇒ (a² + 2 – a) (a² + 2 + a)

Question 14) 4x⁴ + 1

Solution:

4x⁴ + 4x² + 1 – 4x²

⇒ [(2x²)² + 2 × 2x² × 1 + (1)²] – (2x)²

⇒ (2x² +1)² – (2x)²

⇒ (2x² + 1 – 2x) (2x² + 1 + 2x)

Question 15) 4x⁴+ y⁴

Solution:

4x⁴ + 4x²y² + y⁴ – 4x²y²

⇒ [(2x²)²+ 2 × 2x² × y² + (y²)²] – 4x²y²

⇒ (2x² + y²)² – (2xy)²

⇒ (2x² + y² – 2xy) (2x² + y² + 2xy)

Question 16) (x + 2)² – 6(x + 2) + 9

Solution:

(x + 2)² – 6(x + 2) + 9

⇒ (x + 2)² -2 × (x + 2) × 3 + (3)² 

⇒ (x + 2 – 3)²

⇒ (x – 1)²

⇒ (x – 1) (x – 1)

Question 17) 25 – p² – q² – 2pq

Solution:

25 – (p² + q² + 2pq)

⇒ (5)² – (p + q)²

⇒ (5 – p – q) (5 + p + q)

Question 18) x² + 9y² – 6xy – 25a²

Solution:

(x² – 6xy + 9y²) – 25a²

⇒ (x – 3y)² – 25a²

⇒ (x – 3y – 5a) (x – 3y + 5a)

Question 19) 49 – a² + 8ab – 16b²

Solution:

49 – (a² – 8ab + 16b²)

⇒ (7)² – (a- 4b)²

⇒ (7 – a – 4b) (7 + a + 4b)

Question 20) a² – 8ab + 16b² – 25c²

Solution:

a² – 8ab +16b² – 25c²

⇒ (a² – 8ab + 16b²) – 25c²

⇒ (a – 4b)² – (5c)²

⇒ (a – 4b – 5c) (a – 4b + 5c)

Question 21) x² – y² + 6y – 9

Solution:

x² – (y² – 6y + 9)

⇒ x² – (y – 3)²

⇒ (x – y + 3) (x + y – 3)

Question 22) 25x² – 10x + 1 – 36y²

Solution:

25x² – 10x + 1 – 36y²

⇒ [(5x)² – 2 × 5x × 1 + 1²] – 36y²

⇒ (5x – 1)² – (6y)²

⇒ (5x – 1 – 6y) (5x – 1 + 6y)

Question 23) a² – b² + 2bc – c²

Solution:

a² – b² + 2bc – c²

⇒ a² – (b² – 2bc + c²)

⇒ a² – (b – c )²

⇒ (a – b + c) (a + b – c)

Question 24) a² + 2ab + b² – c²

Solution:

a² + 2ab + b² – c²

⇒ (a + b)² – c²

⇒ (a + b – c) (a + b + c)

Question 25) 49 – x²– y² + 2xy

Solution:

49 – x² – y² + 2xy

⇒ 49 – (x² + y² – 2xy)

⇒ (7 – x + y) (7 + x – y)

Question 26) a² + 4b² – 4ab – 4c²

Solution:

a² + 4b² – 4 ab – 4c²

⇒ (a – 2b)² – (2c)²

⇒ (a – 2b – 2c) (a – 2b + 2c)

Question 27) x² – y² – 4xz + 4z²

Solution:

x² – y² – 4xz + 4z²

⇒ (x – 2z)² – y²

⇒ (x – 2z – y)(x – 2z + y)