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Mathematics

Using standard formulae, expand each of the following:

(i) (a + 2b + 3c)2

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

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

Expansions

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Answer

We know that,

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

(i) Given,

⇒ (a + 2b + 3c)2

⇒ (a)2 + (2b)2 + (3c)2 + 2 × (a × 2b + 2b × 3c + 3c × a)

⇒ a2 + 4b2 + 9c2 + 2 × (2ab + 6bc + 3ca)

⇒ a2 + 4b2 + 9c2 + 4ab + 12bc + 6ac

Hence, (a + 2b + 3c)2 = a2 + 4b2 + 9c2 + 4ab + 12bc + 6ac.

(ii) Given,

⇒ (3x + 5y - 2z)2

⇒ [3x + 5y + (-2z)]2

⇒ (3x)2 + (5y)2 + (-2z)2 + 2 × [3x × 5y + 5y × (-2z) + (-2z) × 3x]

⇒ 9x2 + 25y2 + 4z2 + 2 × (15xy - 10yz - 6xz)

⇒ 9x2 + 25y2 + 4z2 + 30xy - 20yz - 12xz

Hence, (3x + 5y - 2z) = 9x2 + 25y2 + 4z2 + 30xy - 20yz - 12xz.

(iii) Given,

⇒ (2x - 3y + 7z)2

⇒ [2x + (-3y) + 7z]2

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

⇒ 4x2 + 9y2 + 49z2 + 2 × [-6xy - 21yz + 14xz]

⇒ 4x2 + 9y2 + 49z2 - 12xy - 42yz + 28xz.

Hence, (2x - 3y + 7z) = 4x2 + 9y2 + 49z2 - 12xy - 42yz + 28xz.

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