Mathematics
For x = 5 and y = 2, verify the following:
(i) (x + y)2 = x2 + 2xy + y2
(ii) (x - y)2 = x2 - 2xy + y2
(iii) x2 - y2 = (x + y)(x - y)
(iv) (x + y)2 = (x - y)2 + 4xy
(v) (x + y)3 = x3 + y3 + 3x2y + 3xy2
Answer
(i) Substituting x = 5 and y = 2 in the given result, we get:
LHS = (x + y)2
= (5 + 2)2
= (7)2
= 49
RHS = x2 + 2xy + y2
= (5)2 + 2(5)(2) + (2)2
= 25 + 20 + 4
= 49
Since, LHS = RHS.
Hence, (x + y)2 = x2 + 2xy + y2 is verified for x = 5 and y = 2.
(ii) Substituting x = 5 and y = 2 in the given result, we get:
LHS = (x - y)2
= (5 - 2)2
= (3)2
= 9
RHS = x2 - 2xy + y2
= (5)2 - 2(5)(2) + (2)2
= 25 - 20 + 4
= 9
Since, LHS = RHS.
Hence, (x - y)2 = x2 - 2xy + y2 is verified for x = 5 and y = 2.
(iii) Substituting x = 5 and y = 2 in the given result, we get:
LHS = x2 - y2
= (5)2 - (2)2
= 25 - 4
= 21
RHS = (x + y)(x - y)
= (5 + 2)(5 - 2)
= 7 × 3
= 21
Since, LHS = RHS.
Hence, x2 - y2 = (x + y)(x - y) is verified for x = 5 and y = 2.
(iv) Substituting x = 5 and y = 2 in the given result, we get:
LHS = (x + y)2
= (5 + 2)2
= (7)2
= 49
RHS = (x - y)2 + 4xy
= (5 - 2)2 + 4(5)(2)
= (3)2 + 40
= 9 + 40
= 49
Since, LHS = RHS.
Hence, (x + y)2 = (x - y)2 + 4xy is verified for x = 5 and y = 2.
(v) Substituting x = 5 and y = 2 in the given result, we get:
LHS = (x + y)3
= (5 + 2)3
= (7)3
= 343
RHS = x3 + y3 + 3x2y + 3xy2
= (5)3 + (2)3 + 3(5)2(2) + 3(5)(2)2
= 125 + 8 + 3(25)(2) + 3(5)(4)
= 125 + 8 + 150 + 60
= 343
Since, LHS = RHS.
Hence, (x + y)3 = x3 + y3 + 3x2y + 3xy2 is verified for x = 5 and y = 2.