Factorise the following, using the identity a² + 2ab + b² = (a + b)²
(i) x² + 6x + 9
(ii) x² + 12x + 36
(iii) x² + 14x + 49
(iv) x² + 2x + 1
(v) 4x² + 4x + 1
(vi) a²x² + 2ax + 1
(vii) a²x² + 2abx + b²
(viii) a²x² + 2abxy + b²y²
(ix) 4x² + 12x + 9
(x) 16x² + 40x + 25
(xi) 9x² + 24x + 16
(xii) 9x² + 30x + 25
(xiii) 2x³ + 24x² + 72x
(xiv) a²x³ + 2abx² + b²x
(xv) 4x⁴ + 12x³ + 9x²
(xvi) (x²/4) + 2x + 4
(xvii) 9x² + 2xy + (y²/9)
Solution:
(i) x² + 6x + 9
Given, x² + 6x + 9
Using the identity a² + 2ab + b² = (a + b)², the expression can be written as,
= (x)² + 2(x)(3) + (3)²
= (x + 3)²
(ii) x² + 12x + 36
Given, x² + 12x + 36
Using the identity a² + 2ab + b² = (a + b)², the expression can be written as,
= (x)² + 2(x)(6) + (6)²
= (x + 6)²
(iii) x² + 14x + 49
Given, x² + 14x + 49
Using the identity a² + 2ab + b² = (a + b)², the expression can be written as,
= (x)² + 2(x)(7) + (7)²
= (x + 7)²
(iv) x² + 2x + 1
Given, x² + 2x + 1
Using the identity a² + 2ab + b² = (a + b)², the expression can be written as,
= (x)² + 2(x)(1) + (1)²
= (x + 1)²
(v) 4x² + 4x + 1
Given, 4x² + 4x + 1
Using the identity a² + 2ab + b² = (a + b)², the expression can be written as,
= (2x)² + 2(2x)(1) + (1)²
= (2x + 1)²
(vi) a²x² + 2ax + 1
Given, a²x² + 2ax + 1
Using the identity a² + 2ab + b² = (a + b)², the expression can be written as,
= (ax)² + 2(ax)(1) + (1)²
= (ax + 1)²
(vii) a²x² + 2abx + b²
Given, a²x² + 2abx + b²
Using the identity a² + 2ab + b² = (a + b)², the expression can be written as,
= (ax)² + 2(ax)(b) + (b)²
= (ax + b)²
(viii) a²x² + 2abxy + b²y²
Given, a²x² + 2abxy + b²y²
Using the identity a² + 2ab + b² = (a + b)², the expression can be written as,
= (ax)² + 2(ax)(by) + (by)²
= (ax + by)²
(ix) 4x² + 12x + 9
Given, 4x² + 12x + 9
Using the identity a² + 2ab + b² = (a + b)², the expression can be written as,
= (2x)² + 2(2x)(3) + (3)²
= (2x + 3)²
(x) 16x² + 40x + 25
Given, 16x² + 40x + 25
Using the identity a² + 2ab + b² = (a + b)², the expression can be written as,
= (4x)² + 2(4x)(5) + (5)²
= (4x + 5)²
(xi) 9x² + 24x + 16
Given, 9x² + 24x + 16
Using the identity a² + 2ab + b² = (a + b)², the expression can be written as,
= (3x)² + 2(3x)(4) + (4)²
= (3x + 4)²
(xii) 9x² + 30x + 25
Given, 9x² + 30x + 25
Using the identity a² + 2ab + b² = (a + b)², the expression can be written as,
= (3x)² + 2(3x)(5) + (5)²
= (3x + 5)²
(xiii) 2x³ + 24x² + 72x
Given, 2x³ + 24x² + 72x = 2x [x² + 12x + 36]
Using the identity a² + 2ab + b² = (a + b)², the expression can be written as,
= 2x [(x)² + 2(x)(6) + (6)² ]
= 2x (x + 6)²
(xiv) a²x³ + 2abx² + b²x
Given, a²x³ + 2abx² + b²x = x [a²x² + 2abx + b²]
Using the identity a² + 2ab + b² = (a + b)², the expression can be written as,
= x [(ax)² + 2(ax)(b) + (b)² ]
= x (ax + b)²
(xv) 4x⁴ + 12x³ + 9x²
Given, 4x⁴ + 12x³ + 9x² = x² (4x² + 12x + 9)
Using the identity a² + 2ab + b² = (a + b)², the expression can be written as,
= x² [(2x)² + 2(2x)(3) + (3)²]
= x² (2x + 3)² = x² (2x + 3)(2x + 3)
(xvi) (x²/4) + 2x + 4
Given, (x²/4) + 2x + 4
Using the identity a² + 2ab + b² = (a + b)², the expression can be written as,
= (x/2)² + 2(x/2)(2) + (2)²
= ((x/2) + 2)²
(xvii) 9x² + 2xy + (y²/9)
Given, 9x² + 2xy + (y²/9)
Using the identity a² + 2ab + b² = (a + b)², the expression can be written as,
= (3x)² + 2(3x)(y/3) + (y/3)²
= (3x + (y/3))²
✦ Try This: Factorise the following, using the identity a² + 2ab + b² = (a + b)²
(i) 49x² + 56x + 16, (ii) 3a³ + 24a² + 48a, (iii) (a²/16) + 4a + 64
☛ Also Check: NCERT Solutions for Class 8 Maths Chapter 9
NCERT Exemplar Class 8 Maths Chapter 7 Problem 89
Factorise the following, using the identity a² + 2ab + b² = (a + b)². (i) x² + 6x + 9, (ii) x² + 12x + 36, (iii) x² + 14x + 49, (iv) x² + 2x + 1, (v) 4x² + 4x + 1, (vi) a²x² + 2ax + 1, (vii) a²x² + 2abx + b², (viii) a²x² + 2abxy + b²y², (ix) 4x² + 12x + 9, (x) 16x² + 40x + 25, (xi) 9x² + 24x + 16, (xii) 9x² + 30x + 25, (xiii) 2x³ + 24x² + 72x, (xiv) a²x³ + 2abx² + b²x, (xv) 4x⁴ + 12x³ + 9x², (xvi) (x²/4) + 2x + 4, (xvii) 9x² + 2xy + (y²/9)
Summary:
Factorising the following, using the identity a² + 2ab + b² = (a + b)²; (i) x² + 6x + 9, (ii) x² + 12x + 36, (iii) x² + 14x + 49, (iv) x² + 2x + 1, (v) 4x² + 4x + 1, (vi) a²x² + 2ax + 1, (vii) a²x² + 2abx + b², (viii) a²x² + 2abxy + b²y², (ix) 4x² + 12x + 9, (x) 16x² + 40x + 25, (xi) 9x² + 24x + 16, (xii) 9x² + 30x + 25, (xiii) 2x³ + 24x² + 72x, (xiv) a²x³ + 2abx² + b²x, (xv) 4x⁴ + 12x³ + 9x², (xvi) (x²/4) + 2x + 4, (xvii) 9x² + 2xy + (y²/9), we get, (x + 3)², (x + 6)², (x + 7)², (x + 1)², (2x + 1)², (ax + 1)², (ax + b)², (ax + by)², (2x + 3)², (4x + 5)², (3x + 4)², (3x + 5)², 2x (x + 6)², x(ax + b)² , x² (2x + 3)(2x + 3), ((x/2) + 2)², (3x + (y/3))² respectively
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