Factorise the following, using the identity (a - b)2 = a2 - 2ab +b2.
(i) x2 - 8x + 16
(ii) x2 - 10x + 25
(iii) y2 - 14y + 49
(iv) p2 - 2p + 1
(v) 4a2 - 4ab + b2
(vi) p2y2- 2py + 1
(vii) a2y2 - 2aby + b2
(viii) 9x2 - 12x + 4
(ix) 4y2 - 12y + 9
(x) x2/4 - 2x + 4
(xi) a2y3 - 2aby2 + b2y
(xii) 9y2 - 4xy + 4x2/9
Solution:
(i) x2 - 8x + 16
Given, x2 - 8x + 16
Using the identity: (a - b)2 = a2 - 2ab + b2, the expression can be written as,
x2 - 8x + 16 = (x)2 - 2(x)(4) + (4)2
= (x - 4)2
(ii) x2 - 10x + 25
Given, x2 - 10x + 25
Using the identity: (a - b)2 = a2 - 2ab + b2, the expression can be written as,
x2 - 10x + 25 = (x)2 - 2(x)(5) + (5)2
= (x - 5)2
(iii) y2 - 14y + 49
Given, y2 - 14y + 49
Using the identity: (a - b)2 = a2 - 2ab + b2, the expression can be written as,
y2 - 14y + 49 = (y)2 - 2(y)(7) + (7)2
= (y - 7)2
(iv) p2 - 2p + 1
Given, p2 - 2p + 1
Using the identity: (a - b)2 = a2 - 2ab + b2, the expression can be written as,
p2 - 2p + 1 = (p)2 - 2(p)(1) + (1)2
= (p - 1)2
(v) 4a2 - 4ab + b2
Given, 4a2 - 4ab + b2
Using the identity: (a - b)2 = a2 - 2ab + b2, the expression can be written as,
4a2 - 4ab + b2 = (2a)2 - 2(2a)(b) + (b)2
= (2a - b)2
(vi) p2y2- 2py + 1
Given, p2y2- 2py + 1
Using the identity: (a - b)2 = a2 - 2ab + b2, the expression can be written as,
p2y2- 2py + 1 = (py)2 - 2(py)(1) + (1)2
= (py - 1)2
(vii) a2y2 - 2aby + b2
Given, a2y2 - 2aby + b2
Using the identity: (a - b)2 = a2 - 2ab + b2, the expression can be written as,
a2y2 - 2aby + b2 = (ay)2 - 2(ay)(b) + (b)2
= (ay - b)2
(viii) 9x2 - 12x + 4
Given, 9x2 - 12x + 4
Using the identity: (a - b)2 = a2 - 2ab + b2, the expression can be written as,
9x2 - 12x + 4 = (3x)2 - 2(3x)(2) + (2)2
= (3x - 2)2
(ix) 4y2 - 12y + 9
Given, 4y2 - 12y + 9
Using the identity: (a - b)2 = a2 - 2ab + b2, the expression can be written as,
4y2 - 12y + 9 = (2y)2 - 2(2y)(3) + (3)2
= (2y - 3)2
(x) x2/4 - 2x + 4
Given, x2/4 - 2x + 4
Using the identity: (a - b)2 = a2 - 2ab + b2, the expression can be written as,
x2/4 - 2x + 4 = (x/2)2 - 2(x/2)(2) + (2)2
= [(x/2) - 2] 2
(xi) a2y3 - 2aby2 + b2y
Given, a2y3 - 2aby2 + b2y = y [a2y2 - 2aby + b2]
Using the identity: (a - b)2 = a2 - 2ab + b2, the expression can be written as,
y [a2y2 - 2aby + b2] = y [(ay)2 - 2(ay)(b) + (b)2]
= y (ay - b)2
(xii) 9y2 - 4xy + 4x2/9
Given, 9y2 - 4xy + 4x2/9
Using the identity: (a - b)2 = a2 - 2ab + b2, the expression can be written as,
9y2 - 4xy + 4x2/9 = (3y)2 - 2(3y)(2x/3) + (2x/3)2
= [3y - (2x/3)]2
✦ Try This:Factorise the following, using the identity (a - b)2 = a2 - 2ab + b2
(i) a2 - 24a + 144, (ii) a6y9 - 4a³y5 + 4y, (iii) 9x² - (12xy/5) + (4y²/25)
☛ Also Check: NCERT Solutions for Class 8 Maths Chapter 9
NCERT Exemplar Class 8 Maths Chapter 7 Problem 90
Factorise the following, using the identity (a - b)2 = a2 - 2ab +b2. (i) x2 - 8x + 16, (ii) x2 - 10x + 25, (iii) y2 - 14y + 49, (iv) p2 - 2p + 1, (v) 4a2 - 4ab + b2, (vi) p2y2- 2py + 1, (vii) a2y2 - 2aby + b2, (viii) 9x2 - 12x + 4, (ix) 4y2 - 12y + 9, (x) x2/4 - 2x + 4, (xi) a2y3 - 2aby2 + b2y, (xii) 9y2 - 4xy + 4x2/9
Summary: Factorising the following, using the identity (a - b)2 = a2 - 2ab +b2, (i) x2 - 8x + 16 , (ii) x2 - 10x + 25, (iii) y2 - 14y + 49 , (iv) p2 - 2p + 1, (v) 4a2 - 4ab + b2, (vi) p2y2- 2py + 1, (vii) a2y2 - 2aby + b2, (viii) 9x2 - 12x + 4, (ix) 4y2 - 12y + 9, (x) x2/4 - 2x + 4 (xi) a2y3 - 2aby2 + b2y (xii) 9y2 - 4xy + 4x2/9, we get,(x - 4)2, (x - 5)2, (y - 7)2, (p - 1)2, (2a - b)2, (py - 1)2, (ay - b)2, (3x - 2)2, (2y - 3)2, [(x/2) - 2] 2, y (ay - b)2, [3y - (2x/3)]2
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