ML Aggarwal Solution Class 9 Chapter 4 Factorisation Exercise 4.3

Exercise 4.3

Question 1

(i) 4x2-25y2

∵a2-b2=(a+b)(a-b)

⇒(2x)2-(5y)2

⇒(2x+5y)(2x-5y)


(ii) 9x2-1

⇒(3x)2-12

⇒(3x+1)(3x-1)


Question 2

(i) 150-6a2

⇒6(25-a2)

⇒6(52-a2)

⇒6(5+a)(5-a)


(ii) 32x2-18y2

⇒2(16x2-9y2)

⇒2[(4x)2-(3y)2]

⇒2(4x+3y)(4x-3y)


Question 3

(i) (x-y)2-9

⇒(x-y)2-32

⇒(x-y+3)(x-y-3)


(ii) 9(x+y)2-x2

⇒9[(x+y)2-x2]

⇒9[(x+y+x)(x+y-x)]

⇒9(2x+y)y

⇒9y(2x+y)


Question 4

(i) 20x2-45y2

⇒5(4x2-9y2)

⇒5((2x)2-(3y)2)

⇒5(2x+3y)(2x-3y)


(ii) 9x2-4(y+2x)2

⇒(3x)2-(2(y+2x))2

⇒(3x+2y+4x)(3x-2y-4x)

⇒(7x+2y)(-x-2y)

⇒-(7x-2y)(x+y)


Question 5

(i) 2(x-2y)2-50y2

⇒2[(x-2y)2-25y2]

⇒2[(x-2y)2-(5y)2]

⇒2[(x-2y+5y)(x-2y-5y)]

⇒(x+3y)(x-7y)

⇒2(x+3y)(x-7y)


(ii) 32-2(x-4)2

⇒2[16-(x-4)2]

⇒2[42-(x-4)2]

⇒2[(4+x-4)(4-x+4)]

⇒2.x.(8-x)

⇒2x(8-x)


Question 6

(i) 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

⇒πa[a4-π2b2]

⇒πa[(a2)2-(πb)2]

⇒πa[(a2+πb)(a2-πb)]


Question 7

(i) 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(6x-2)(4x+2)


(ii) (x-2)(x+2)+3

⇒(x2-22)+3

⇒x2-4+3

⇒x2-1

⇒x2-12

⇒(x+1)(x-1)


Question 8

(i) x-2y-x2+4y2

⇒x-2y-(x2-4y2)

⇒x-2y-(x2-(2y)2)

⇒(x-2y)-[(x+2y)(x-2y)]

⇒(x-2y)(1-(x+2y))

⇒(x-2y)(1-x-2y)


(ii) 4a2-b2+2a+b

⇒(2a)2-b2+2a+b

⇒((2a+b)(2a-b)+1(2a+b))

⇒(2a+b)(2a-b+1)


Question 9

(i) 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)

⇒a2-a-b2+b

⇒a2-b2-a+b

⇒(a+b)(a-b)-1(a-b)

⇒(a-b)(a+b-1)


Question 10

(i) 9-x2+2xy-y2

⇒9-x2+xy+xy-y2

⇒9-x2+xy+3x-3x+3y-3y+xy-y2

⇒9-3x+3y+3x-x2+xy+xy-3y-y2

⇒3(3-x+y)+x(3-x+y)+y(-3-y+x)

⇒3(3-x+y)+x(3-x+y)-y(3-x+y)

⇒(3+x-y)(3-x+y)


(ii) 9x4-(x2+2x+1)

⇒9x4-x2-2x+1

⇒9x4-3x3+3x3-3x2-x2-x-x+1

⇒9x4-3x3-3x2+3x3-x2-x+3x2-x-1

⇒3x2(3x2-x-1)+x(3x2-x-1)+1(3x2-x-1)

⇒(3x2-x-1)(3x2+x+1)


Question 11

(i) 9x4-x2-12x-36

⇒9x4-3x3+3x3-18x2+18x2-x2-12x-36

⇒9x4-3x3-18x2+3x3-x2-6x+18x2-6x-36

⇒3x2(3x2-x-16)+x(3x2-x-6)++6(3x2-x-6)

⇒(3x2-x-16)(3x2+x+6)


(ii) x3-5x2-x+5

⇒x3-x-5x2+5

⇒x(x2-1)-5(x2-1)

⇒(x2-1)(x-5)

⇒(x2-12)(x-5)

⇒(x+1)(x-1)(x-5)


Question 12

(i) a4-b4+2b2-1

⇒a4-b4-a2b2+a2-a2+b2+b2-1

⇒a4-a2b2+a2+a2b2-b4+b2-a2+b2-1

⇒a2(a2-b2+1)+b2(a2-b2+1)-1(a2-b2+1)

⇒(a2-b2+1)(a2+b2-1)


(ii) x3-25x

⇒x(x2-25)

⇒x(x2-52)

⇒x(x+5)(x-5)


Question 13

(i) 2x4-32

⇒2(x4-16)

⇒2(x4-24)

⇒2[(x2)2-(22)2]

⇒2(x2+4)(x2-4)


(ii) a2(b+c)-(b+c)3

⇒(b+c)(a2-(b+c)2)

⇒(b+c)(a+(b+c))(a-(b+c))

⇒(b+c)(a+b+c)(a-b-c)


Question 14

(i) (a+b)3-a-b

⇒(a+b)3-(a+b)

⇒(a+b)[(a+b)2-12]

⇒(a+b)(a+b+1)(a+b-1)


(ii) x2-2xy+y2-a2-2ab-b2

⇒(x+y)2-(a2+2ab+b2)

⇒(x-y)2-(a+b)2

⇒(x-y+a+b)(x-y-a-b)


Question 15

(i) (a2-b2)(c2-d2)-4abcd

⇒a2(c2-d2)-b2(c2-d2)-4abcd

⇒a2c2-a2d2-b2c2+b2d2-4abcd

⇒a2c2+b2d2-a2d2-b2c2-2abcd-2abcd

⇒a2c2+b2d2-2abcd-a2d2-b2c2-2abcd

⇒(ac-bd)2-(ad-bc)2

⇒(ac-bd+ad-bc)(ac-bd-ad+bc)


(ii) 4x2-y2-3xy+2x-2y

⇒x2+3x2-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


Question 16

(i) $x^2+\frac{1}{x^2}-11$

⇒$x^2+\frac{1}{x^2}-2-9$

⇒$\left(x^2+\frac{1}{x^2}-2\right)-3^2$

⇒$\left(x-\frac{1}{2}\right)^2-3^2$

⇒$\left(x-\frac{1}{x}+3\right)\left(x-\frac{1}{x}-3\right)$


(ii) x4+5x2+9

⇒x4+5x2+x2-x2+32

⇒(x2)2+6x2+32-x2

⇒(x2+3)2-x2

⇒(x2+3+x)(x2+3-x)


Question 17

(i) a4+b4-7a2b2

⇒a4+b4+2a2b2-7a2b2

⇒(a2)2+(b2)2+2.a2b2-9a2b2

⇒(a2+b2)2-(3ab)2

⇒(a2+b2+3ab)(a2+b2-3ab)


Question 18

(i) (x2-5x+7)(x2+5x+7)

⇒[(x2+7)-5x][(x2+7)+5x]

⇒(x2+7)2-(5x)2

⇒(x2+7)2-25x2


(ii) (x2-5x+7)(x2-5x-7)

⇒[(x2-5x)+7][(x2-5x)-7]

⇒(x2-5x)2-72

⇒(x2-5x)2-49


(iii) (x2+5x-7)(x2-5x+7)

⇒(x2+(5x-7))(x2-(5x-7))

⇒x2-(5x-7)2

⇒x2-(25x2+49-70x)

⇒x2-25x2-49+70x

⇒-24x2+70x-49


Question 19

(i) (979)2-(21)2

⇒(979+21)(979-21)

⇒(1000)(958)

⇒958000


(ii) (99.9)2-(0.1)2

⇒(99.9+0.1)(99.9-0.1)

⇒(100)(99.8)

⇒9980

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