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Expand each of the following, using suitable identities:
i) (x + 2y + 4z)² ii) (2x - y + z)² iii) (-2x + 3y + 2z)² iv) (3a - 7b - c)²
v) (-2x + 5y - 3z)² vi) [(1/4)a - (1/2)b + 1]²

Solution:

i) (x + 2y + 4z)²

Using the identity: (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

Taking a = x, b = 2y, c = 4z

(x + 2y + 4z)² = x² + (2y)² + (4z)² + 2(x)(2y) + 2(2y)(4z) + 2(4z)(x)

= x² + 4y² + 16z² + 4xy + 16yz + 8zx

ii) (2x - y + z)²

Using the identity: (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

Taking a = 2x, b = -y, c = z

(2x - y + z)² = (2x)² + (-y)² + (z)² + 2(2x)(-y) + 2(-y)(z) + 2(z)(2x)

= 4x² + y² + z² - 4xy - 2yz + 4zx

iii) (-2x + 3y + 2z)²

Using the identity: (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

Taking a = -2x, b = 3y, c = 2z

(-2x + 3y + 2z)² = (-2x)² + (3y)² + (2z)² + 2(-2x)(3y) + 2(3y)(2z) + 2(2z)(-2x)

= 4x² + 9y² + 4z² - 12xy + 12yz - 8zx

iv) (3a - 7b - c)²

Using the identity: (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

Taking a = 3a, b = -7b, c = -c

(3a - 7b - c)² = (3a)² + (-7b)² + (-c)² + 2(3a)(-7b) + 2(-7b)(-c) + 2(-c)(3a)

= 9a² + 49b² + c² - 42ab + 14bc - 6ca

v) (-2x + 5y - 3z)²

Using the identity: (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

Taking a = -2x, b = 5y, c = -3z

(-2x + 5y - 3z)² = (-2x)² + (5y)² + (-3z)² + 2(-2x)(5y) + 2(5y)(-3z) + 2(-3z)(-2x)

= 4x² + 25y² + 9z² - 20xy - 30yz + 12zx

vi) [(1/4)a - (1/2)b + 1]²

Using the identity: (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

Taking, a = (1/4)a, b = (-1/2)b, c = 1

[(1/4)a - (1/2)b + 1]² = [(a/4)² + (-b/2)² + (1)² + 2(a/4)(-b/2) + 2(-b/2)(1) + 2(1)(a/4)]

= (1/16)a² + (1/4)b² + 1 - (1/4)ab - b + (1/2)a

☛ Check: NCERT Solutions CBSE Class 9 Maths Chapter 2

Video Solution:

Expand each of the following, using suitable identities: i) (x + 2y + 4z)² ii) (2x - y + z)² iii) (-2x + 3y + 2z)² iv) (3a - 7b - c)² v) (-2x + 5y - 3z)² vi) [(1/4)a - (1/2)b + 1]²

NCERT Solutions Class 9 Maths Chapter 2 Exercise 2.5 Question 4

Summary:

The expanded form of each of the following (x + 2y + 4z)², (2x - y + z)², (-2x + 3y + 2z)², (3a - 7b - c)², (-2x + 5y - 3z)², and [(1/4)a - (1/2)b + 1]² using suitable identities are x² + 4y² + 16z² + 4xy + 16yz + 8zx, 4x² + y² + z² - 4xy - 2yz + 4zx, 4x² + 9y² + 4z² - 12xy + 12yz - 8zx, 9a² + 49b² + c² - 42ab + 14bc - 6ca, 4x² + 25y² + 9z² - 20xy - 30yz + 12zx, and (1/16)a² + (1/4)b² + 1 - (1/4)ab - b + (1/2)a respectively.

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