Using standard formulae, expand each of the following: (i) (a + 2b + 3c)2 (ii) (3x + 5y - 2z)2 (iii) (2x - 3y + 7z)2

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.

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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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

Answer

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