Class 8 Maths Solution | Lesson 10 Factorization | Curriculum Development Centre (CDC)

Factorization of the expressions of the form ax²+ bx + c

Exercise 10.1.3

1. Complete the perfect square by filling the appropriate term in the blank space:

(a) x² +….+ 16

= (x)² + 2 x x x 4 + (4)²

= (x)² + 8x + (4)²

= 8x

(b) 4a² + .... + y²

= 4a² + 2 x a x y + y² = 2ay

(c) p² - ..... + 36

= (p)² - 2 x p x 6 + (6)² = 2pq

(d) 9a² - .... + 16b²

= (3a)² - 2 x 3a x 4b + (4b)² = 24ab

(e) 25p² - ..... + 49q²

= (5p)² - 2 x 5p x 7q + (7q)² = 70pq (f) p² + ..... + 4/p²

= (p)² + 2 × p × 2/p + (2/p)²

= 4

(g) 225x² - ..... + 64y²

= (15x)² - 2 × 15x × 8y + (8y)²

= 240xy

(h) 1 + ..... + 36y²

= (1)² + 2 × 1 × 6y + (6y)²

= 12y

(i) p² - ..... + 1/p²

= (p)² - 2 × p × 1/p + (1/p)²

= 2

Exercise 10.1.4

2. Factorize:

(a) a² + 12a + 36

= (a)² + 2 × a × 6 + (6)²

= (a + 6)²

(b) y² + 14y + 49

= (y)² + 2 × y × 7 + (7)²

= (y + 7)²

(c) p² + 22p + 121

= (p)² + 2 × p × 11 + (11)²

= (p + 11)²

(d) 4a² + 20a + 25

= (2a)² + 2 × 2a × 5 + (5)²

= (2a + 5)²

(e) 9r² + 60r + 100

= (3r)² + 2 × 3r × 10 + (10)²

= (3r + 10)²

(f) 36x² + 84x + 49

= (6x)² + 2 × 6x × 7 + (7)²

= (6x + 7)²

(g) x² – 8x + 16

= (x)² - 2 × x × 4 + (4)²

= (x - 4)²

(h) a² – 18a + 81

= (a)² - 2 × a × 9 + (9)²

= (a - 9)²

(i) p² – 26p + 169

= (p)² - 2 × p × 13 + (13)²

= (p - 13)²

(j) 9a² – 30a + 25

= (3a)² - 2 × 3a × 5 + (5)²

= (3a - 5)²

(k) 25y² – 60y + 36

= (5y)² - 2 × 5y × 6 + (6)²

= (5y - 6)²

(l) 49r² – 70r + 25

= (7r)² - 2 × 7r × 5 + (5)²

= (7r - 5)²

(m) 4p² + 24pq + 36q²

= (2p)² + 2 × 2p × 6q + (6q)²

= (2p + 6q)²

(n) 9a² + 42ab + 49b²

= (3a)² + 2 × 3a × 7b + (7b)²

= (3a + 7b)²

(o) x²/16 – xy + 4y²

= (x/4)² - 2 × (x/4) × 2y + (2y)²

= (x/4 - 2y)²

(p) 25a² – 40ab + 16b²

= (5a)² - 2 × 5a × 4b + (4b)²

= (5a - 4b)²

(q) 49q² – 70qr + 25r²

= (7q)² - 2 × 7q × 5r + (5r)²

= (7q - 5r)²

(r) 25x² – 2xy + y²/25

= (5x)² - 2 × 5x × (y/5) + (y/5)²

= (5x - y/5)²

3. Factorize:

(a) a² + 12a + 36 – b²

= (a² + 12a + 36) - b²

= (a + 6)² - b²

= (a + 6 + b)(a + 6 - b)

(b) y² + 16y + 64 – z²

= (y² + 16y + 64) - z²

= (y + 8)² - z²

= (y + 8 + z)(y + 8 - z)

(c) p² + 26p + 169 – 9q²

= (p² + 26p + 169) - 9q²

= (p + 13)² - (3q)²

= (p + 13 + 3q)(p + 13 - 3q)

(d) 4a² – b² + 20b – 100

= (4a² - 100) + (20b - b²)

= (2a - 10)(2a + 10) - (b² - 20b)

= (2a - 10)(2a + 10) - b(b - 20)

= (2a - 10 - b)(2a + 10 + b)

(e) 9r² – s² – 6s – 9

= (9r² - 9) - (s² + 6s)

= 9(r² - 1) - (s² + 6s)

= 9(r - 1)(r + 1) - s(s + 6)

= (9(r - 1) - s)(9(r + 1) + s)

4. Factorize:

(a) x² + 5x + 4

= x² + (4+1)x + 4

= x² + 4x + x + 4

= x(x + 4) + 1(x + 4)

= (x + 4)(x + 1)

(b) x² + 3x + 2

= x² + (2+1)x + 2

= x² + 2x + x + 2

= x(x + 2) + 1(x + 2)

= (x + 2)(x + 1)

(c) x² – 5x + 6

= x² - (3+2)x + 6

= x² - 3x - 2x + 6

= x(x - 3) - 2(x - 3)

= (x - 3)(x - 2)

(d) y² + 5y + 6

= y² + (3+2)y + 6

= y² + 3y + 2y + 6

= y(y + 3) + 2(y + 3)

= (y + 3)(y + 2)

(e) x² + 7x + 12

= x² + (4+3)x + 12

= x² + 4x + 3x + 12

= x(x + 4) + 3(x + 4)

= (x + 4)(x + 3)

(f) a² – 3a + 2

= a² - (2+1)a + 2

= a² - 2a - a + 2

= a(a - 2) - 1(a - 2)

= (a - 2)(a - 1)

(g) a² – 6a + 8

= a² - (4+2)a + 8

= a² - 4a - 2a + 8

= a(a - 4) - 2(a - 4)

= (a - 4)(a - 2)

(h) b² – 5b + 6

= b² - (3+2)b + 6

= b² - 3b - 2b + 6

= b(b - 3) - 2(b - 3)

= (b - 3)(b - 2)

(i) b² + 13b + 42

= b² + (7+6)b + 42

= b² + 7b + 6b + 42

= b(b + 7) + 6(b + 7)

= (b + 7)(b + 6)

(j) b² – 13b + 40

= b² - (8+5)b + 40

= b² - 8b - 5b + 40

= b(b - 8) - 5(b - 8)

= (b - 8)(b - 5)

(k) z² – 13z + 36

= z² - (9+4)z + 36

= z² - 9z - 4z + 36

= z(z - 9) - 4(z - 9)

= (z - 9)(z - 4)

(l) x² – 15x + 56

= x² - (8+7)x + 56

= x² - 8x - 7x + 56

= x(x - 8) - 7(x - 8)

= (x - 8)(x - 7)

(m) x² – 15x + 54

= x² - (9+6)x + 54

= x² - 9x - 6x + 54

= x(x - 9) - 6(x - 9)

= (x - 9)(x - 6)

(n) z² + 15z + 44

= z² + (11+4)z + 44

= z² + 11z + 4z + 44

= z(z + 11) + 4(z + 11)

= (z + 11)(z + 4)

(o) b² – 12b + 36

= b² - (6+6)b + 36

= b² - 6b - 6b + 36

= b(b - 6) - 6(b - 6)

= (b - 6)(b - 6)

(p) b² + 15b + 56

= b² + (8+7)b + 56

= b² + 8b + 7b + 56

= b(b + 8) + 7(b + 8)

= (b + 8)(b + 7)

(q) z² – 12z + 27

= z² - (9+3)z + 27

= z² - 9z - 3z + 27

= z(z - 9) - 3(z - 9)

= (z - 9)(z - 3)

(r) x² – 23x + 102

= x² - (17+6)x + 102

= x² - 17x - 6x + 102

= x(x - 17) - 6(x - 17)

= (x - 17)(x - 6)

(s) (a + b)² + 11(a + b) + 18

= (a + b)² + (9+2)(a + b) + 18

= (a + b)² + 9(a + b) + 2(a + b) + 18

= (a + b)((a + b) + 9) + 2((a + b) + 9)

= ((a + b + 9)((a + b + 2)

(t) (x + y)² – 15(x + y) + 36

= (x + y)² - (12+3)(x + y) + 36

= (x + y)² - 12(x + y) - 3(x + y) + 36

= (x + y)((x + y) - 12) - 3((x + y) - 12)

= ((x + y - 12)((x + y - 3)

4. Factorize:

(a) x² + 4x – 21

= x² + (7 - 3)x - 21

= x² + 7x - 3x - 21

= x(x + 7) - 3(x + 7)

= (x + 7)(x - 3)

(b) x² + x – 20

= x² + (5 - 4)x - 20

= x² + 5x - 4x - 20

= x(x + 5) - 4(x + 5)

= (x + 5)(x - 4)

(c) x² + 3x – 28

= x² + (7 - 4)x - 28

= x² + 7x - 4x - 28

= x(x + 7) - 4(x + 7)

= (x + 7)(x - 4)

(d) y² – 6y – 27

= y² + (3 - 9)y - 27

= y² + 3y - 9y - 27

= y(y + 3) - 9(y + 3)

= (y + 3)(y - 9)

(e) x² + 7x – 18

= x² + (9 - 2)x - 18

= x² + 9x - 2x - 18

= x(x + 9) - 2(x + 9)

= (x + 9)(x - 2)

(f) a² + 10a – 39

= a² + (13 - 3)a - 39

= a² + 13a - 3a - 39

= a(a + 13) - 3(a + 13)

= (a + 13)(a - 3)

(g) a² – a – 132

= a² + (11 - 12)a - 132

= a² + 11a - 12a - 132

= a(a + 11) - 12(a + 11)

= (a + 11)(a - 12)

(h) b² – 8b – 65

= b² - (13 + 5)b - 65

= b² - 13b + 5b - 65

= b(b - 13) + 5(b - 13)

= (b - 13)(b + 5)

(i) b² + 3b – 108

= b² + (12 - 9)b - 108

= b² + 12b - 9b - 108

= b(b + 12) - 9(b + 12)

= (b + 12)(b - 9)

(j) b² – 7b – 120

= b² - (15 + 8)b - 120

= b² - 15b + 8b - 120

= b(b - 15) + 8(b - 15)

= (b - 15)(b + 8)

(k) z² – 29z – 132

= z² - (33 + 4)z - 132

= z² - 33z + 4z - 132

= z(z - 33) + 4(z - 33)

= (z - 33)(z + 4)

(l) x² + xy – 240y²

= x² + (16y - 15y)x - 240y²

= x² + 16xy - 15xy - 240y²

= x(x + 16y) - 15y(x + 16y)

= (x + 16y)(x - 15y)

(m) 35 – 2x – x²

= -x² - 2x + 35

= -(x² + 2x - 35)

= -{x² + (7 - 5)x - 35}

= -{x² + 7x - 5x - 35}

= -{x(x + 7)- 5(x + 7)}

= -(x + 7) (x - 5)}

(n) 96 – 4z – z²

= -z² - 4z + 96

= -(z² + 4z - 96)

= -{z² + (12 - 8) z - 96}

= -{(z² + 12z -8z - 96)

= -{z(z + 12) -8z(z + 12)

= -{(z + 12)(z - 8z)

(o) 72 + b – b²

= -b² + b + 72

= -(b² - b - 72)

= -(b² - 9b + 8b - 72)

= -{b(b - 9) + 8(b - 9)}

= -(b - 9)(b + 8)

(p) (a + b)² + 5(a + b) – 36

= (a + b)² + (9 - 4)(a + b) - 36

= {(a + b)² + 9(a + b) - 4(a + b) - 36}

= {(a + b)((a + b) + 9) - 4((a + b) + 9)}

= ((a + b + 9)((a + b - 4)

(q) (x + y)² – 9(x + y) – 112

= (x + y)² - (16 + 7)(x + y) - 112

= {(x + y)² - 16(x + y) + 7(x + y) - 112}

= {(x + y)((x + y) - 16) + 7((x + y) - 16)}

= ((x + y - 16)((x + y + 7)

(a) 3x² + 5x + 2

= 3x² + (3+2)x + 2

= 3x² + 3x + 2x + 2

= 3x (x + 1) + 2(x + 1)

= (x + 1)(3x + 2)

(b) 3x² - 4x + 1

= 3x² - (3+1)x + 1

= 3x² - 3x - x + 1

= 3x(x - 1) - 1(x - 1)

= (x - 1)(3x - 1)

(c) 7x² - 30x + 8

= 7x² - (28 + 2)x + 8

= 7x² - 28x - 2x + 8

= 7x(x - 4) - 2(x - 4)

= (x - 4)(7x - 2)

(d) 4a² - 8a + 3

= 4a² - (6+2)a + 3

= 4a² - 6a - 2a + 3

= 2a(2a - 3) - 1(2a - 3)

= (2a - 3)(2a - 1)

(e) 15p² - 13p + 2

= 15p² - (10 + 3)p + 2

= 15p² - 10p - 3p + 2

= 5p(3p - 2) - 1(3p - 2)

= (3p - 2)(5p - 1)

(f) 12a² - 32a + 5

= 12a² - (30 + 2)a + 5

= 12a² - 30a - 2a + 5

= 6a(2a - 5) - 1(2a - 5)

= (2a - 5)(6a - 1)

(g) 5x² - 14x - 3

= 5x² - (15 - 1)x - 3

= 5x² - 15x + x - 3

= 5x(x - 3) + 1(x - 3)

= (x - 3)(5x + 1)

(h) 10x² - 3x - 1

= 10x² - (5 - 2)x - 1

= 10x² - 5x + 2x - 1

= 5x(2x - 1) + 1(2x - 1)

= (2x - 1)(5x + 1)

(i) 15p² - 13p + 2

= 15p² - (10 + 3)p + 2

= 15p² - 10p - 3p + 2

= 5p(3p - 2) - 1(3p - 2)

= (3p - 2)(5p - 1)

(j) 6b² - 4b - 10

= 6b² - (10 - 6)b - 10

= 6b² - 10b + 6b - 10

= 2b(3b - 5) + 2(3b - 5)

= (3b - 5)(2b + 2)

(k) 21x² + 25x + 4

= 21x² + (21 + 4)x + 4

= 21x² + 21x + 4x + 4

= 21x(x + 1) + 4(x + 1)

= (x + 1)(21x + 1)

(l) 12a² + 28ab - 5b²

= 12a² + (30 - 2)ab - 5b²

= 12a² + 30ab - 2ab - 5b²

= 6a(2a + 5b) - b(2a + 5b)

= (2a + 5b)(6a - b)

(m) 16a² + 24ab + 9b²

= 16a² + (12 + 12)ab + 9b²

= 16a² + 12ab + 12ab + 9b²

= 4a(4a + 3b) + 3b(4a + 3b)

= (4a + 3b)(4a + 3b)

(n) 6x² + xy - 7y²

= 6x² + (7 - 6)xy - 7y²

= 6x² + 7xy - 6xy - 7y²

= x(6x + y) - y(6x + 7y)

= (6x + y)(x - y)

(o) 3a² - ab - 10b²

= 3a² - (6 - 5)ab - 10b²

= 3a² - 6ab + 5ab - 10b²

= 3a(a - 2b) + 5b(a - 2b)

= (a - 2b)(3a + 5b)

(p) 6p²q + 30pq + 36q

= 6q(p² + 5p + 6)

= 6q{p² + (3 + 2)p + 6)}

= 6q{p² + 3p + 2p + 6)}

= 6q{p(p + 3) + 2 (p + 3)}

= 6q{(p + 3) (p + 2)}

(q) 6a² + 35ab - 6b²

= 6a² + (36 - 1)ab - 6b²

= 6a² + 36ab - ab - 6b²

= 6a(a + 6b) - b(a + 6b)

= (a + 6b)(6a - b)

(r) 6a² - 5ab - 6b²

= 6a² - (6 - 1)ab - 6b²

= 6a² - 6ab + ab - 6b²

= 3a(2a - b) + b(a - 6b)

= (2a - b)(3a + b)

(s) 4 + 17x - 15x²

= -15x² + 17x + 4

= -(15x² - 17x - 4)

= -(15x² - 20x + 3x - 4)

= -(5x(3x - 4) + 1(3x - 4))

= -(3x - 4)(5x + 1)

(t) 6 - 13a + 6a²

= 6a² - 13a + 6

= 6a² - 9a - 4a + 6

= 3a(2a - 3) - 2(2a - 3)

= (2a - 3)(3a - 2)