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# 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 = 16m^{2}/9 + 9n^{2}/16

(iv) (4pq + 3q)^{2 } - (4pq - 3q)^{2} = 48pq^{2}

(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} = a^{2} + 2ab + b^{2}]

= 9x^{2} + 42x + 49 - 84x

= 9x^{2} - 42x + 49

R.H.S = (3x - 7)^{2}

= (3x)^{2} - 2(3x)(7) + (7)^{2} [Since, (a - b)^{2} = a^{2} - 2ab + b^{2}]

= 9x^{2} - 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} = a^{2} - 2ab + b^{2}]

= 81p^{2} - 90pq + 25q^{2} + 180pq

= 81p^{2} + 90pq + 25q^{2}

R.H.S = (9p + 5q)^{2}

= (9p)^{2} + 2(9p)(5q) + (5q)^{2} [Since, (a + b)^{2} = a^{2} + 2ab + b^{2}]

= 81p^{2} + 90pq + 25q^{2}

L.H .S = R.H .S

(iii) (4m/3 - 3n/4)^{2} + 2mn = 16m^{2}/9 + 9n^{2}/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} = a^{2} - 2ab + b^{2}]

= 16m^{2}/9 - 2mn + 9n^{2}/16 + 2mn

= 16m^{2}/9 + 9n^{2}/16

L.H.S = R.H.S

(iv) (4pq + 3q)^{2 } - (4pq - 3q)^{2} = 48pq^{2}

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} = a^{2} + 2ab + b^{2 }and (a - b)^{2} = a^{2} - 2ab + b^{2}]

= 16p^{2}q^{2} + 24pq^{2} + 9q^{2} - [16p^{2}q^{2} - 24pq^{2} + 9q^{2}]

= 16p^{2}q^{2 }+ 24pq^{2} + 9q^{2 }- 16p^{2}q^{2 }+ 24 pq^{2} - 9q^{2}

= 48pq^{2}

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)

= (a^{2} - b^{2}) + (b^{2} - c^{2}) + (c^{2} - a^{2}) [Since, a^{2} - b^{2} = (a - b)(a + b)]

= a^{2} - b^{2} + b^{2} - c^{2} + c^{2} - a^{2}

= 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 = 16m^{2}/9 + 9n^{2}/16 (iv) (4pq + 3q)^{2 } - (4pq - 3q)^{2} = 48pq^{2 }(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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