Show that
(i)(3x + 7)2 - 84x = (3x - 7)2
(ii) (9p - 5q)2 + 180pq = (9p + 5q)2
(iii) (4m/3 - 3n/4)2 + 2mn = 16m2/9 + 9n2/16
(iv) (4pq + 3q)2 - (4pq - 3q)2 = 48pq2
(v) (a - b)(a + b) + (b - c)(b + c) + (c - a)(c + a) = 0
Solution:
(i) (3x + 7)2 - 84x = (3x - 7)2
L.H.S = (3x + 7)2 - 84x
= (3x)2 + 2(3x)(7) + (7)2 - 84x [Using algebraic identity, (a + b)2 = a2 + 2ab + b2]
= 9x2 + 42x + 49 - 84x
= 9x2 - 42x + 49
R.H.S = (3x - 7)2
= (3x)2 - 2(3x)(7) + (7)2 [Since, (a - b)2 = a2 - 2ab + b2]
= 9x2 - 42x + 49
L.H .S = R.H .S
(ii) (9p - 5q)2 + 180pq = (9p + 5q)2
L.H .S = (9p - 5q)2 + 180pq
= (9p)2 - 2(9p)(5q) + (5q)2 +180pq [Since, (a - b)2 = a2 - 2ab + b2]
= 81p2 - 90pq + 25q2 + 180pq
= 81p2 + 90pq + 25q2
R.H.S = (9p + 5q)2
= (9p)2 + 2(9p)(5q) + (5q)2 [Since, (a + b)2 = a2 + 2ab + b2]
= 81p2 + 90pq + 25q2
L.H .S = R.H .S
(iii) (4m/3 - 3n/4)2 + 2mn = 16m2/9 + 9n2/16
L.H.S = (4m/3 - 3n/4)2 + 2mn
= (4m/3)2 - 2(4m/3)(3n/4) + (3n/4)2 + 2mn [Since, (a - b)2 = a2 - 2ab + b2]
= 16m2/9 - 2mn + 9n2/16 + 2mn
= 16m2/9 + 9n2/16
L.H.S = R.H.S
(iv) (4pq + 3q)2 - (4pq - 3q)2 = 48pq2
L.H.S = (4pq + 3q)2 - (4pq - 3q)2
= (4pq)2 + 2(4pq)(3q) + (3q)2 - [(4pq)2 - 2(4pq)(3q) + (3q)2] [Since, (a + b)2 = a2 + 2ab + b2 and (a - b)2 = a2 - 2ab + b2]
= 16p2q2 + 24pq2 + 9q2 - [16p2q2 - 24pq2 + 9q2]
= 16p2q2 + 24pq2 + 9q2 - 16p2q2 + 24 pq2 - 9q2
= 48pq2
L.H.S = R.H.S
(v) (a - b)(a + b) + (b - c)(b + c) + (c - a)(c + a) = 0
L.H.S = (a - b)(a + b) + (b - c)(b + c) + (c - a)(c + a)
= (a2 - b2) + (b2 - c2) + (c2 - a2) [Since, a2 - b2 = (a - b)(a + b)]
= a2 - b2 + b2 - c2 + c2 - a2
= 0
L.H.S = R.H.S
☛ Check: NCERT Solutions for Class 8 Maths Chapter 9
Video Solution:
Show that (i)(3x + 7)² - 84x = (3x - 7)² (ii) (9p - 5q)² + 180pq = (9p + 5q)² (iii) (4m/3 - 3n/4)² + 2mn = 16m²/9 + 9n²/16 (iv) (4pq + 3q)² - (4pq - 3q)² = 48pq² (v) (a - b)(a + b) + (b - c)(b + c) + (c - a)(c + a) = 0
NCERT Solutions Class 8 Maths Chapter 9 Exercise 9.5 Question 5
Summary:
We have proved the following (i)(3x + 7)2 - 84x = (3x - 7)2 (ii) (9p - 5q)2 + 180pq = (9p + 5q)2 (iii) (4m/3 - 3n/4)2 + 2mn = 16m2/9 + 9n2/16 (iv) (4pq + 3q)2 - (4pq - 3q)2 = 48pq2 (v) (a - b)(a + b) + (b - c)(b + c) + (c - a)(c + a) = 0
☛ Related Questions:
- Use a suitable identity to get each of the following products. (i) (x + 3)(x + 3) (ii) (2y + 5)(2y + 5) (iii) (2a - 7)(2a - 7) (iv) (3a - (1/2))(3a - (1/2)) (v) (1.1 m - 0.4)(1.1 m + 0.4) (vi)(a2 + b2)(-a2 + b2) (vii) (6x - 7)(6x + 7) (viii) (-a + c)(-a + c) (ix) (x/2 + 3y/4)(x/2 + 3y/4) (x) (7a - 9b)(7a - 9b)
- Use the identity (x + a)(x + b) = x2 + (a + b)x + ab to find the following products. (i) (x + 3)(x + 7) (ii) (4x + 5)(4x + 1) (iii) (4x - 5)(4x -1) (iv) (4x + 5)(4x -1) (v) (2x + 5y)(2x + 3 y) (vi) (2a2 + 9)(2a2 + 5) (vii) (xyz - 4)(xyz - 2)
- Find the following squares by using the identities. (i)(b - 7)2 (ii) (xy + 3z)2 (iii) (6x2 - 5 y)2 (iv) (2m/3 + 3n/2)2 (v) (0.4 p - 0.5q)2 (vi) (2xy + 5 y)2
- 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
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