Question

Write the following in the expanded form:

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

(ii) (2a − 3b − c)2

Answer 1

(i) (a + 2b + c)2 = a2 + (2b)2 + c2 + 2a(2b) + 2ac + 2(2b)c = a2 + 4b2 + c2 + 4ab + 2ac + 4bc (ii) (2a − 3b − c)2 = [(2a) + (−3b) + (−c)]2 = (2a)2 + (−3b)2 + (−c)2 + 2(2a)(−3b) + 2(−3b)(−c) + 2(2a)(−c) = 4a2 + 9b2 + c2 − 12ab + 6bc − 4ca (iii) (−3x+y+z)2 = [(−3x)2 + y2 + z2 + 2(−3x)y + 2yz + 2(−3x)z = 9x2 + y2 + z2 − 6xy + 2yz − 6xz (iv) (m+2n−5p)2 = m2 + (2n)2 + (−5p)2 + 2m × 2n + (2×2n×−5p) + 2m × −5p = m2 + 4n2 + 25p2 + 4mn − 20np − 10pm (v) (2+x−2y)2 = 22 + x2 + (−2y)2 + 2(2)(x) + 2(x)(−2y) + 2(2)(−2y) = 4 + x2 + 4y2 + 4 x − 4xy − 8y (vi) (a2 +b2 +c2)2 = (a2)2 + (b2)2 + (c2)2 + 2a2 b2 + 2b2c2 + 2a2c2 = a4 + b4 + c4 + 2a2 b2 + 2b2 c2 + 2c2 a2 (vii) (ab + bc + ca)2 = (ab)2 + (bc)2 + (ca)2 + 2(ab)(bc) + 2(bc)(ca) + 2(ab)(ca) = a2 b2 + b2c2 + c2 a2 + 2(ac)b2 + 2(ab)(c)2 + 2(bc)(a)2 (viii) (xy+yz+zx)2(xy+yz+zx)2 (ix) (abc+bac+cab)2(abc+bac+cab)2 (x) (x + 2y + 4z)2 = x2 + (2y)2 + (4z)2 + (2x)(2y) + 2(2y)(4z) + 2x(4z) = x2 + 4y2 + 16z2 + 4xy + 16yz + 8xz (xi) (2x − y + z)2 = (2x)2 + (−y)2 + (z)2 + 2(2x)(−y) + 2(−y)(z) + 2(2x)(z) = 4x2 + y2 + z2 − 4xy − 2yz + 4xz (xii) (−2x + 3y + 2z)2 = (−2x)2 + (3y)2 + ( 2z)2 + 2(−2x)(3y) +2(3y)(2z) +2(−2x)(2z) = 4x2 + 9y2 + 4z2 −12xy + 12yz − 8xzRead more on Sarthaks.com - https://www.sarthaks.com/604166/write-the-following-in-the-expanded-form-i-a-2b-c-2-ii-2a-3b-c-2?show=604187#a604187

Answer 2

Answer:

(i) (a + 2b + c)2  = a2 + (2b)2 + c2 + 2a(2b) + 2ac + 2(2b)c  = a2 + 4b2 + c2 + 4ab + 2ac + 4bc 

(ii) (2a − 3b − c)2  = [(2a) + (−3b) + (−c)]2  = (2a)2 + (−3b)2 + (−c)2 + 2(2a)(−3b) + 2(−3b)(−c) + 2(2a)(−c)  = 4a2 + 9b2 + c2 − 12ab + 6bc − 4