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

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

**Solution:**

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

= (x + 3) (x + 3)

= (x + 3)^{2}

= x^{2 }+ 6x + 9

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

**Solution:**

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

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

= (2y + 5)^{2}

= 4y^{2} + 20y + 25

**(iii) (2a – 7) (2a – 7)**

**Solution:**

Using (a-b) ^{2 }= a^{2} + b^{2} – 2ab

= (2a – 7) (2a – 7) = (2a – 7)^{2}

= 4a^{2} – 28a + 49

**(iv) (3a – 1/2) (3a – 1/2)**

**Solution:**

Using (a-b) ^{2} = a^{2 }+ b^{2} – 2ab

= (3a – 1/2) (3a – 1/2)

= (3a – 1/2)^{2}

= 9a^{2} -3a+(1/4)

**(v) (1.1m – 0.4) (1.1m + 0.4)**

**Solution:**

Using (a – b) (a + b) = a^{2} – b^{2}

(1.1m – 0.4) (1.1m + 0.4)

= 1.21m^{2} – 0.16

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

**Solution:**

Using (a – b) (a + b) = a^{2} – b^{2}

= (a^{2}+ b^{2}) (– a^{2}+ b^{2})

= (b^{2} + a^{2}) (b^{2} – a^{2})

= -a^{4} + b^{4}

**(vii) (6x – 7) (6x + 7)**

**Solution:**

Using (a – b) (a + b) = a^{2} – b^{2}

= (6x – 7) (6x + 7)

=36x^{2} – 49

**(viii) (- a + c) (- a + c)**

**Solution:**

Using (a-b) ^{2} = a^{2} + b^{2} – 2ab

= (– a + c) (– a + c) = (– a + c)^{2}

= c^{2} + a^{2 }– 2ac

**(ix) (1/2x + 3/4y) (1/2x + 3/4y)**

**Solution:**

Using (a-b) ^{2} = a^{2} + b^{2} – 2ab

= (7a – 9b) (7a – 9b) = (7a – 9b)^{2}

= 49a^{2} – 126ab + 81b^{2}

(**x) (7a – 9b) (7a – 9b)**

**Solution:**

Using (a – b)^{2} = a^{2} – b^{2} + 2ab

= (7a – 9b)^{2}

= 49a^{2} + 81b^{2} – 126ab

**2. Use the identity (x + a) (x + b) = x2 + (a + b) x + ab to find the following products.**

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

**Solution:**

= (x + 3) (x + 7)

= x^{2 }+ (3+7)x + 21

= x^{2} + 10x + 21

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

**Solution:**

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

= 16x^{2} + 4x + 20x + 5

= 16x^{2} + 24x + 5

**(iii) (4x – 5) (4x – 1)**

**Solution:**

= (4x – 5) (4x – 1)

= 16x^{2} – 4x – 20x + 5

= 16x^{2} – 24x + 5

**(iv) (4x + 5) (4x – 1)**

**Solution:**

= (4x + 5) (4x – 1)

= 16x^{2} + (5-1)4x – 5

= 16x^{2} +16x – 5

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

**Solution:**

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

= 4x^{2} + (5y + 3y)2x + 15y^{2}

= 4x^{2} + 16xy + 15y^{2}

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

**Solution:**

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

= 4a4 + (9+5)2a^{2} + 45

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

**(vii) (xyz – 4) (xyz – 2)**

**Solution:**

= (xyz – 4) (xyz – 2)

= x^{2}y^{2}z^{2} + (-4 -2)xyz + 8

= x^{2}y^{2}z^{2} – 6xyz + 8

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

In this question we Use these identities:

(a – b) ^{2} = a^{2} + b^{2} – 2ab

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

**(i) (b – 7) ^{2}**

**Solution:**

= (b – 7)^{2}

= b^{2} – 14b + 49

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

**Solution:**

= (xy + 3z)^{2}

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

**(iii) (6x ^{2 }– 5y)^{2}**

**Solution:**

= (6x^{2} – 5y)^{2}

= 36x^{4} – 60x^{2}y + 25y^{2}

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

**Solution:**

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

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

**(v) (0.4p – 0.5q) ^{2}**

**Solution:**

= (0.4p – 0.5q)2

= 0.16p^{2} – 0.4pq + 0.25q^{2}

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

**Solution:**

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

= 4x^{2}y^{2} + 20xy^{2} + 25y^{2}

**4. Simplify.**

**(i) (a ^{2 }– b2) ^{2}**

**Solution:**

= a^{4} + b^{4} – 2a^{2}b^{2}

**(ii) (2x + 5) ^{2 }– (2x – 5)^{2}**

**Solution:**

= 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:**

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

= 98m^{2} + 128n^{2}

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

**Solution:**

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

= 41m2 + 80mn + 41n^{2}

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

**Solution:**

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

= 4p^{2 }– 4q^{2}

**(vi) (ab + bc) ^{2}– 2ab²c**

**Solution:**

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

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

**5. Show that.**

**(i) (3x + 7) ^{2} – 84x = (3x – 7)^{2}**

**Solution:**

Let us take LHS:

LHS = (3x + 7)^{2 }– 84x

= 9x^{2} + 42x + 49 – 84x

= 9x^{2} – 42x + 49

= (3x – 7)^{2}

Hence,

LHS = RHS

**(ii) (9p – 5q) ^{2}+ 180pq = (9p + 5q)^{2}**

**Solution:**

Let us take LHS:

LHS = (9p – 5q)^{2}+ 180pq

= 81p^{2} – 90pq + 25q^{2} + 180pq

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

Now, taking RHS:

RHS = (9p + 5q)^{2}

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

Hence, LHS = RHS

**(iii) (4/3m – 3/4n) ^{2} + 2mn = 16/9 m^{2 }+ 9/16 n^{2}**

**Solution:**

Let us take LHS:

LHS = (4/3m – 3/4n)^{2} + 2mn

= 16/9 m^{2} + 9/16 n^{2} – 2mn + 2mn

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

Hence, LHS = RHS

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

**Solution:**

Let us take LHS:

LHS = (4pq + 3q)^{2}– (4pq – 3q)^{2}

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

= 48pq^{2}

Hence,

LHS = RHS

**(v) (a – b) (a + b) + (b – c) (b + c) + (c – a) (c + a) = 0**

**Solution:**

Let us take LHS:

LHS = (a – b) (a + b) + (b – c) (b + c) + (c – a) (c + a)

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

= 0

Hence,

LHS = RHS

**6. Using identities, evaluate.**

**(i) 71²**

**Solution:**

According to question:

= (70+1)^{2}

= 70^{2} + 140 + 1^{2}

= 4900 + 140 +1

= 5041

**(ii) 99²**

**Solution:**

According to question:

= (100 -1)^{2}

= 100^{2 }– 200 + 1^{2}

= 10000 – 200 + 1

= 9801

**(iii) 102 ^{2}**

**Solution:**

According to question:

= (100 + 2)^{2}

= 100^{2 }+ 400 + 2^{2}

= 10000 + 400 + 4 = 10404

**(iv) 998²**

**Solution:**

According to question:

= (1000 – 2)^{2}

= 1000^{2} – 4000 + 2^{2}

= 1000000 – 4000 + 4

= 996004

**(v) 5.2²**

**Solution:**

According to question:

= (5 + 0.2)^{2}

= 5^{2} + 2 + 0.2^{2}

= 25 + 2 + 0.04

= 27.04

**(vi) 297 x 303**

**Solution:**

According to question:

= (300 – 3) (300 + 3)

= 300^{2} – 3^{2}

= 90000 – 9

= 89991

**(vii) 78 x 82**

**Solution:**

According to question:

= (80 – 2) (80 + 2)

= 80^{2} – 2^{2}

= 6400 – 4

= 6396

**(viii) 8.9 ^{2}**

**Solution:**

According to question:

= (9 – 0.1)^{2}

= 9^{2} – 1.8 + 0.1^{2}

= 81 – 1.8 + 0.01

= 79.21

**(ix) 10.5 x 9.5**

**Solution:**

According to question:

= (10 + 0.5) (10 – 0.5)

= 10^{2} – 0.5^{2}

= 100 – 0.25

= 99.75

**7. Using a ^{2} – b^{2} = (a + b) (a – b), find**

**(i) 51 ^{2}– 49^{2}**

**Solution:**

According to question:

= (51 + 49) (51 – 49)

= 100 x 2 = 200

**(ii) (1.02) ^{2}– (0.98)^{2}**

**Solution:**

According to question:

= (1.02 + 0.98) (1.02 – 0.98)

= 2 x 0.04

= 0.08

**(iii) 153 ^{2}– 147^{2}**

**Solution:**

According to question:

= (153 + 147) (153 – 147)

= 300 x 6

= 1800

**(iv) 12.1 ^{2}– 7.9^{2}**

**Solution:**

According to question:

= (12.1 + 7.9) (12.1 – 7.9)

= 20 x 4.2

= 84

**8. Using (x + a) (x + b) = x ^{2} + (a + b) x + ab, find**

**(i) 103 x 104**

**Solution:**

According to question:

= (100 + 3) (100 + 4)

= 100^{2} + (3 + 4)100 + 12

= 10000 + 700 + 12 = 10712

**(ii) 5.1 x 5.2**

**Solution:**

According to question:

= (5 + 0.1) (5 + 0.2)

= 5^{2} + (0.1 + 0.2)5 + 0.1 x 0.2

= 25 + 1.5 + 0.02

= 26.52

**(iii) 103 x 98**

**Solution:**

According to question:

= (100 + 3) (100 – 2)

= 100^{2} + (3-2)100 – 6

= 10000 + 100 – 6

= 10094

**(iv) 9.7 x 9.8**

**Solution:**

According to question:

= (9 + 0.7) (9 + 0.8)

= 9^{2} + (0.7 + 0.8)9 + 0.56

= 81 + 13.5 + 0.56

= 95.06

**👍👍👍**