Page 102 - Start Up Mathematics_8 (Non CCE)

2
= p + pq + pq + q 2
2
2
= p + 2pq + q (Commutative property qp = pq)
2
2
\ (p + q) = p + 2pq + q 2

2
2
II. (p – q) = p – 2pq + q 2
In other words,
2
2
(Difference of the two terms) = (First term) – 2 ¥ (First term) ¥ (Second term) + (Second term) 2
2
Proof: (p – q) = (p – q)(p – q)
= p(p – q) – q(p – q) (Distributive property of multiplication over subtraction)
2
= p – pq – qp + q 2
2
= p – 2pq + q 2 (Commutative property qp = pq)
2
2
\ (p – q) = p – 2pq + q 2

2

III. (p + q) (p – q) = p – q 2
In other words,
2
(Sum of the two terms) ¥ (Difference of the two terms) = (First term) – (Second term) 2
Proof: (p + q)(p – q) = p(p – q) + q(p – q) (Distributive property of multiplication over addition)
2
2
= p – pq + pq – q (Commutative property qp = pq)
2
= p – q 2
2
\ (p + q)(p – q) = p – q 2

Example 24: Find the following squares by using the identities: (NCERT)

2
2
n
(a) (6x – 5y) (b) Ê 2 m + 3 ˆ 2
Á
˜
Ë
2 ¯
3
Ê
2
(c) (0.4p – 0.5q) (d) Ê -6 p 3 + q 3 ˆ -6 p 3 - q 3 ˆ
Á
˜
˜ Á
¯
Ë 7
¯ Ë 7
2
2
2
2
2
2
2
2 2
Solution: (a) (6x – 5y) = (6x ) – 2(6x )(5y) + (5y) {Using (p – q) = p – 2pq + q }
2
4
= 36x – 60x y + 25y 2
Ê 2 3 ˆ 2 Ê 2 ˆ 2 Ê 2 ˆ Ê 3 ˆ Ê 3 ˆ 2 2 2 2
n +
m
(b) Á 3 m + 2 ¯ = Á Ë 3 ¯ ˜ + 2 Á 3 ¯ Ë 2 ¯ Á Ë 2 ¯ {Using (p + q) = p + 2pq + q }
n
n
m
˜
˜
˜
˜ Á
Ë
Ë
4 2 9 2
= m + 2mn + n
9 4
2
2
2
2
2
2
(c) (0.4p – 0.5q) = (0.4p) – 2(0.4p)(0.5q) + (0.5q) {Using (p – q) = p – 2pq + q }
2
= 0.16p – 0.4pq + 0.25q 2
Ê -6 ˆ -6 ˆ Ê -6 ˆ 2 2 2
Ê
32
(d) Á p 3 + q 3 ˜ Á p 3 - q 3 ˜ = Á Ë 7 p 3 ˜ ¯ - q() {Using (p + q)(p – q) = p – q }
¯
¯ Ë 7
Ë 7
36
6
= p - q 6
49
94