**Question 1. Use a suitable identity to get each of the following products.**

**(i) (x + 3) (x + 3) **

**Solution:**

(x + 3) (x + 3)

Putting formula (a + b)

^{2 }= a^{2 }+ b^{2}+ 2abPut a = x & b = 3

(x + 3) (x +3 ) = (x + 3)

^{2}= x

^{2 }+ 6x + 9

**(ii) (2y + 5) (2y + 5)**

**Solution:**

(2y + 5) (2y + 5)

Putting formula (a + b)

^{2 }= a^{2 }+ b^{2 }+ 2abPut a = 2y & b = 5

(2y + 5) (2y + 5) = (2y + 5)

^{2}= 4y

^{2 }+ 20y + 25

**(iii) (2a – 7) (2a – 7)**

**Solution:**

(2a – 7) (2a – 7)

Putting formula (x – y)

^{2 }= x^{2 }+ y^{2 }– 2xyPut x = 2a & y = 7

(2a – 7) (2a – 7) = (2a – 7)

^{2}= 4a

^{2 }– 28a + 49

**(iv) (3a – 1/2) (3a – 1/2)**

**Solution:**

(3a – 1/2) (3a – 1/2)

Putting formula (x – y)

^{2}= x^{2}+ y^{2}– 2xyPut x = 3a & y = 1/2

= (3a)

^{2}+ (1/2)^{2}– 2 (3a) (1/2)= 3

^{2}a^{2}+ 1/4 – 3a= 9a

^{2}+ 1/4 – 3a

**(v) (1.1m – 0.4) (1.1m + 0.4)**

**Solution:**

(1.1m – 0.4) (1.1m + 0.4)

Putting formula (a + b) (a – b) = a

^{2}– b^{2}Put a = 1.1m & b = 0.4

(1.1m – 0.4) (1.1m + 0.4) = (1.1m)

^{2 }– (0.4)^{2}= 1.21m

^{2 }– 0.16

**(vi) (a ^{2} + b^{2}) (– a^{2} + b^{2})**

**Solution:**

(a

^{2}+ b^{2}) (– a^{2}+ b^{2})(a

^{2}+ b^{2}) (– a^{2}+ b^{2}) = (b^{2}+ a^{2}) (b^{2 }– a^{2})Putting formula (x + y) (x – y) = x

^{2 }– y^{2}Put x = b

^{2}& y = a^{2}(b

^{2}+ a^{2}) (b^{2 }– a^{2}) = b^{2×2}– a^{2×2}= b

^{4}– a^{4}

**(vii) (6x – 7) (6x + 7)**

**Solution:**

(6x – 7) (6x + 7)

Putting formula (a – b) (a + b) = a

^{2 }– b^{2}Put a = 6x & b = 7

(6x – 7) (6x + 7) = (6x)

^{2}– 7^{2}= 36x

^{2}– 49

**(viii) (– a + c) (– a + c)**

**Solution:**

(– a + c) (– a + c)

(– a + c) (– a + c) = (c – a) (c – a)

Putting formula (x – y)

^{2 }= x^{2 }+ y^{2 }– 2xyPut x = c & y = a

(c – a) (c – a) = c

^{2}+ a^{2}– 2ca= a

^{2}+ c^{2}– 2ac

**(ix) (x/2 + 3y/4) (x/2 + 3y/4)**

**Solution:**

(x/2 + 3y/4) (x/2 + 3y/4)

Putting formula (a + b)

^{2}= a^{2}+ b^{2}+ 2abput a = x/2 & b = 3y/4

= (x/2)

^{2}+ (3y/4)^{2}+ 2 (x/2) (3y/4)= x

^{2}/4 + 9y^{2}/16 + 3xy/4

**(x) (7a – 9b) (7a – 9b)**

**Solution:**

(7a – 9b) (7a – 9b)

Putting formula (x – y)

^{2 }= x^{2}+ y^{2}– 2xyPut x = 7a & y = 9b

(7a – 9b) (7a – 9b) = (7a)

^{2}+ (9b)^{2}– 2(7a)(9b)= 49a

^{2}+ 81b^{2}– 126ab

**Question 2. Use the identity (x + a) (x + b) = x**^{2} + (a + b) x + ab to find the following products.

^{2}+ (a + b) x + ab to find the following products.

**(i) (x + 3) (x + 7)**

**Solution:**

(x + 3) (x + 7)

Formula (x + a) (x + b) = x

^{2}+ (a + b) x + abPut a = 3 & b = 7

= x

^{2}+ (3 + 7) x + (3 * 7)= x

^{2}+10x + 21

**(ii) (4x + 5) (4x + 1)**

**Solution:**

(4x + 5) (4x + 1)

Formula (y + a) (y + b) = y

^{2}+ (a + b) y + abPut y = 4x , a = 5 & b = 1

= (4x)

^{2}+ (5 + 1) 4x + (5 * 1)= 16x

^{2}+ 24x + 5

**(iii) (4x – 5) (4x – 1)**

**Solution:**

(4x – 5) (4x – 1)

Formula (y + a) (y + b) = y

^{2}+ (a + b) y + abPut y = 4x , a = -5 & b = -1

= (4x)

^{2}+ (-5 – 1) 4x + (-5 * -1)= 16x

^{2}– 24x + 5

**(iv) (4x + 5) (4x – 1)**

**Solution:**

(4x + 5) (4x – 1)

Formula (y + a) (y + b) = y

^{2}+ (a + b) y + abPut y = 4x , a = 5 & b = -1

= (4x)

^{2}+ (5 – 1) 4x + (5 * -1)= 16x

^{2}+ 16x – 5

**(v) (2x + 5y) (2x + 3y)**

**Solution:**

(2x + 5y) (2x + 3y)

Formula (t + a) (t + b) = t

^{2}+ (a + b) t + abPut t = 2x , a = 5y & b = 3y

= (2x)

^{2}+ ( 5y + 3y) 2x + (5y * 3y)= 4x

^{2}+ 16xy + 15y^{2}

**(vi) (2a ^{2 }+ 9) (2a^{2} + 5)**

**Solution:**

(2a

^{2}+ 9) (2a^{2}+ 5)Formula (x + y) (x + z) = x

^{2}+ (y + z) x + yzPut x = 2a

^{2}, y = 9 & z = 5= (2a

^{2})^{2}+ (9 + 5) 2a^{2}+ (9 * 5)= 4a

^{4}+ 28a^{2}+ 45

**(vii) (xyz – 4) (xyz – 2)**

**Solution:**

(xyz – 4) (xyz – 2)

Formula (t + a) (t + b) = t

^{2}+ (a + b) t + abPut t = xyz , a = -4 & b = -2

= (xyz)

^{2}+ (-4 + (-2)) xyz + ((-4) * (-2))= x

^{2}y^{2}z^{2}– 6xyz + 8

**Question 3. Find the following squares by using the identities.**

**(i) (b – 7) ^{2}**

**Solution:**

(b – 7)

^{2}Using Formula (x – y)

^{2}= x^{2}+ y^{2}– 2xyPutting x = b & y = 7

= b

^{2}+ 72 – 2(b)(7)= b

^{2}– 14b + 49

**(ii) (xy + 3z) ^{2}**

**Solution:**

(xy + 3z)

^{2}Using Formula (a + b)

^{2}= a^{2 }+ b^{2}+ 2abPutting a = xy & b = 3z

= x

^{2}y^{2}+ 6xyz + 9z^{2}

**(iii) (6x ^{2} – 5y)^{2}**

**Solution:**

(6x

^{2}– 5y)^{2}Using Formula (a – b)

^{ 2}= a^{2}+ b^{2}– 2abPutting a = 6x

^{2}& b = 5y= 36x

^{4}– 60x^{2}y + 25y^{2}

**(iv) [(2m/3) + (3n/2)] ^{2}**

**Solution:**

[(2m/3) + (3n/2)]

^{2}Using Formula (a + b)

^{2}= a^{2}+ b^{2}+ 2abPutting a = 2m/3 & b = 3n/2

= (2m/3)

^{2}+ (3n/2)^{2}+ 2 (2m/3) (3n/2)= (4m

^{2}/9) + (9n^{2}/4) + 2mn

**(v) (0.4p – 0.5q) ^{2}**

**Solution:**

(0.4p – 0.5q)

^{2}Using Formula (a – b)

^{2}= a^{2}+ b^{2}– 2abPutting a = 0.4p & b = 0.5q

= 0.16p

^{2}– 0.4pq + 0.25q^{2}

**(vi) (2xy + 5y) ^{2}**

**Solution:**

(2xy + 5y)

^{2}Using Formula (a + b)

^{2}= a^{2}+ b^{2}+ 2abPutting a = 2xy & b = 5y

= (2xy)

^{2}+ (5y)^{2}+ 2 (2xy) (5y)= 4x

^{2}y^{2}+ 20xy^{2}+ 25y^{2}

**Question 4. Simplify.**

**(i) (a ^{2} – b^{2})^{2}**

**Solution:**

(a

^{2}– b^{2})^{2}Putting formula (x – y)

^{2}= x^{2}+ y^{2}– 2xyPut x = a

^{2}& y = b^{2}= a

^{4}+ b^{4}– 2a^{2}b^{2}

**(ii) (2x + 5) ^{2} – (2x – 5)^{2}**

**Solution:**

(2x + 5)

^{2}– (2x – 5)^{2}Putting formula (a + b)

^{2}= a^{2}+ b^{2}+ 2ab & (a – b)^{2}= a^{2}+ b^{2}– 2ab= 4x

^{2}+ 20x + 25 – (4x^{2}– 20x + 25)= 4x

^{2}+ 20x + 25 – 4x^{2}+ 20x – 25= 40x

**(iii) (7m – 8n) ^{2} + (7m + 8n)^{2}**

**Solution:**

(7m – 8n)

^{2}+ (7m + 8n)^{2}Putting formula (a – b)

^{2}= a^{2}+ b^{2}– 2ab & (a + b)^{2}= a^{2}+ b^{2}+ 2ab= (49m

^{2}– 112mn + 64n^{2}) + (49m^{2}+ 112mn + 49n^{2})= 98m

^{2}+ 128n^{2}

**(iv) (4m + 5n) ^{2} + (5m + 4n)^{2}**

**Solution:**

(4m + 5n)

^{2}+ (5m + 4n)^{2}Putting formula (a + b)

^{2}= a^{2}+ b^{2}+ 2ab= (16m

^{2}+ 40mn + 25n^{2}) + (25m^{2}+ 40mn + 16n^{2})= 41m

^{2}+ 80mn + 41n^{2}

**(v) (2.5p – 1.5q) ^{2} – (1.5p – 2.5q)^{2}**

**Solution:**

(2.5p – 1.5q)

^{2}– (1.5p – 2.5q)^{2}Putting formula (a – b)

^{2}= a^{2}+ b^{2}– 2ab= (6.25p

^{2}– 7.5pq + 2.25q^{2}) – (2.25p^{2}+ 7.5pq – 6.25q^{2})= 4p

^{2}– 4q^{2}

**(vi) (ab + bc) ^{2} – 2ab^{2}c**

**Solution:**

(ab + bc)

^{2 }– 2ab²cPutting formula (a + b)

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

^{2}b^{2}+ 2ab^{2}c + b^{2}c^{2}) – 2ab^{2}c= a

^{2}b^{2}+ b^{2}c^{2}

**(vii) (m ^{2} – n^{2}m)^{2} + 2m^{3}n^{2}**

**Solution:**

(m

^{2}– n^{2}m)^{2}+ 2m^{3}n^{2}Putting formula (a – b)

^{2}= a^{2}+ b^{2}– 2ab= (m

^{4}– 2m^{3}n^{2}+ m^{2}n^{4}) + 2m^{3}n^{2}= m

^{4}+ m^{2}n^{4}

### Chapter 9 Algebraic Expressions and Identities – Exercise 9.5 | **Set 2**

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