Simplify.
(i) (a2
− b2)2 (ii) (2x +5)2
− (2x − 5)2
(iii) (7m −
8n)2 + (7m + 8n)2
(iv) (4m + 5n)2 + (5m + 4n)2
(v) (2.5p −
1.5q)2 − (1.5p − 2.5q)2
(vi) (ab + bc)2
− 2ab2c (vii) (m2 −
n2m)2 + 2m3n2
Simplify.
(i) (a2 − b2)2 (ii) (2x +5)2 − (2x − 5)2
(iii) (7m − 8n)2 + (7m + 8n)2 (iv) (4m + 5n)2 + (5m + 4n)2
(v) (2.5p − 1.5q)2 − (1.5p − 2.5q)2
(vi) (ab + bc)2 − 2ab2c (vii) (m2 − n2m)2 + 2m3n2
(i) (a2
− b2)2 = (a2)2
− 2(a2) (b2) + (b2)2
[(a − b)2 = a2 −
2ab + b2 ]
=
a4 − 2a2b2
+ b4
(ii) (2x +5)2
− (2x − 5)2 = (2x)2 +
2(2x) (5) + (5)2 − [(2x)2 −
2(2x) (5) + (5)2]
[(a − b)2 = a2 −
2ab + b2]
[(a + b)2 = a2 + 2ab
+ b2]
= 4x2 + 20x + 25 − [4x2
− 20x + 25]
= 4x2 + 20x + 25 − 4x2
+ 20x − 25 = 40x
(iii) (7m −
8n)2 + (7m + 8n)2
= (7m)2 − 2(7m) (8n) + (8n)2
+ (7m)2 + 2(7m) (8n) + (8n)2
[(a
− b)2 = a2 − 2ab
+ b2 and (a + b)2 =
a2 + 2ab + b2]
=
49m2 − 112mn + 64n2
+ 49m2 + 112mn + 64n2
=
98m2 + 128n2
(iv) (4m + 5n)2
+ (5m + 4n)2
=
(4m)2 + 2(4m) (5n) + (5n)2
+ (5m)2 + 2(5m) (4n) + (4n)2
[ (a + b)2 = a2 + 2ab
+ b2]
=
16m2 + 40mn + 25n2 + 25m2
+ 40mn + 16n2
=
41m2 + 80mn + 41n2
(v) (2.5p −
1.5q)2 − (1.5p − 2.5q)2
=
(2.5p)2 − 2(2.5p) (1.5q) +
(1.5q)2 − [(1.5p)2 −
2(1.5p)(2.5q) + (2.5q)2]
[(a
− b)2 = a2 − 2ab
+ b2 ]
=
6.25p2 − 7.5pq + 2.25q2
− [2.25p2 − 7.5pq + 6.25q2]
=
6.25p2 − 7.5pq + 2.25q2
− 2.25p2 + 7.5pq − 6.25q2]
=
4p2 − 4q2
(vi) (ab + bc)2
− 2ab2c
=
(ab)2 + 2(ab)(bc) + (bc)2
− 2ab2c [(a + b)2
= a2 + 2ab + b2 ]
=
a2b2 + 2ab2c
+ b2c2 − 2ab2c
=
a2b2 + b2c2
(vii) (m2
− n2m)2 + 2m3n2
=
(m2)2 − 2(m2)
(n2m) + (n2m)2
+ 2m3n2 [(a − b)2
= a2 − 2ab + b2 ]
=
m4 − 2m3n2
+ n4m2 + 2m3n2
=
m4 + n4m2
(i) (a2 − b2)2 = (a2)2 − 2(a2) (b2) + (b2)2 [(a − b)2 = a2 − 2ab + b2 ]
= a4 − 2a2b2 + b4
(ii) (2x +5)2 − (2x − 5)2 = (2x)2 + 2(2x) (5) + (5)2 − [(2x)2 − 2(2x) (5) + (5)2]
[(a − b)2 = a2 − 2ab + b2]
[(a + b)2 = a2 + 2ab + b2]
= 4x2 + 20x + 25 − [4x2 − 20x + 25]
= 4x2 + 20x + 25 − 4x2 + 20x − 25 = 40x
(iii) (7m − 8n)2 + (7m + 8n)2
= (7m)2 − 2(7m) (8n) + (8n)2 + (7m)2 + 2(7m) (8n) + (8n)2
[(a − b)2 = a2 − 2ab + b2 and (a + b)2 = a2 + 2ab + b2]
= 49m2 − 112mn + 64n2 + 49m2 + 112mn + 64n2
= 98m2 + 128n2
(iv) (4m + 5n)2 + (5m + 4n)2
= (4m)2 + 2(4m) (5n) + (5n)2 + (5m)2 + 2(5m) (4n) + (4n)2
[ (a + b)2 = a2 + 2ab + b2]
= 16m2 + 40mn + 25n2 + 25m2 + 40mn + 16n2
= 41m2 + 80mn + 41n2
(v) (2.5p − 1.5q)2 − (1.5p − 2.5q)2
= (2.5p)2 − 2(2.5p) (1.5q) + (1.5q)2 − [(1.5p)2 − 2(1.5p)(2.5q) + (2.5q)2]
[(a − b)2 = a2 − 2ab + b2 ]
= 6.25p2 − 7.5pq + 2.25q2 − [2.25p2 − 7.5pq + 6.25q2]
= 6.25p2 − 7.5pq + 2.25q2 − 2.25p2 + 7.5pq − 6.25q2]
= 4p2 − 4q2
(vi) (ab + bc)2 − 2ab2c
= (ab)2 + 2(ab)(bc) + (bc)2 − 2ab2c [(a + b)2 = a2 + 2ab + b2 ]
= a2b2 + 2ab2c + b2c2 − 2ab2c
= a2b2 + b2c2
(vii) (m2 − n2m)2 + 2m3n2
= (m2)2 − 2(m2) (n2m) + (n2m)2 + 2m3n2 [(a − b)2 = a2 − 2ab + b2 ]
= m4 − 2m3n2 + n4m2 + 2m3n2
= m4 + n4m2
