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Mathematics

Using standard formulae, expand each of the following:

(i) (6 - 2y + 4z)2

(ii) (4x - 3y + z)2

(iii) (7 - 2x - 3y)2

Expansions

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Answer

We know that,

⇒ (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca).

(i) Given,

⇒ (6 - 2y + 4z)2

⇒ [6 + (-2y) + 4z]2

⇒ (6)2 + (-2y)2 + (4z)2 + 2 × [6 × (-2y) + (-2y) × (4z) + 4z × 6]

⇒ 36 + 4y2 + 16z2 + 2 × [-12y - 8yz + 24z]

⇒ 36 + 4y2 + 16z2 - 24y - 16yz + 48z

Hence, (6 - 2y + 4z)2 = 36 + 4y2 + 16z2 - 24y - 16yz + 48z.

(ii) Given,

⇒ (4x - 3y + z)2

⇒ [4x + (-3y) + z]2

⇒ (4x)2 + (-3y)2 + (z)2 + 2 × [4x × (-3y) + (-3y) × (z) + z × 4x]

⇒ (16x)2 + 9y2 + z2 + 2 × [-12xy - 3yz + 4xz]

⇒ 16x2 + 9y2 + z2 - 24xy - 6yz + 8xz

Hence, (4x - 3y + z)2 = 16x2 + 9y2 + z2 - 24xy - 6yz + 8xz.

(iii) Given,

⇒ (7 - 2x - 3y)2

⇒ [7 + (-2x) + (-3y)]2

⇒ (7)2 + (-2x)2 + (-3y)2 + 2 × [7 × (-2x) + (-2x) × (-3y) + (-3y) × 7]

⇒ 49 + 4x2 + 9y2 + 2 × (-14x + 6xy - 21y)

⇒ 49 + 4x2 + 9y2 - 28x + 12xy - 42y.

Hence, (7 - 2x - 3y)2 = 49 + 4x2 + 9y2 - 28x + 12xy - 42y.

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