# Algebraic Expressions

## Exercise 9.5

### Part 1

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

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

Answer: Using (a + b)^2 = a^2+ 2ab + b^2 we get the following equation:
= x^2 + 6x + 9

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

Answer: 4y^2 + 20y + 25

(iii) (2a – 7) (2a – 7)

Answer: Using (a - b)^2 = a^2 -2ab + b^2 we get the following equation:
= 4a^2 - 28a + 49

(iv) (3a-1/2)(3a-1/2)

Answer: 9a^2-3a+1/4

(v) (1.1m – 0.4) (1.1m + 0.4)

Answer: Using (a - b)(a + b) = a^2 - b^2
= 1.21m^2 - 0.16

(vi) (a^2+ b^2) (– a^2+ b^2)

Answer: = (a^2 - a^2)
= (b^2 + a^2 ) (b^2 - a^2)
= a^4 - b^4

(vii) (6x – 7) (6x + 7)

Answer: 36x^2 - 49

(viii) (– a + c) (– a + c)

Answer: =(c-a)^2=c^2-2ac+a^2

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

Answer: (x^2)/(4)+(9y^2)/(16)+(3xy)/(4)

(x) (7a – 9b) (7a – 9b)

Answer: = 49a^2 - 126ab + 81b^2

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

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

Answer: x^2 + (3+7)x + 21
= x^2 + 10x + 21

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

Answer: = 16x^2 + (5 + 1)4x + 5
= 16x^2 + 24x + 5

(iii) (4x – 5) (4x – 1)

Answer: = 16x^2 + (-5-1)4x + 5
= 16x^2 - 20x + 5

(iv) (4x + 5) (4x – 1)

Answer: = 16x^2 + (5-1)4x - 5
= 16x^2 +16x - 5

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

Answer: = 4x^2 + (5y + 3y)4x + 15y^2
= 4x^2 + 32xy + 15y^2

(vi) (2a^2+ 9) (2a^2+ 5)

Answer: = 4a^4 + (9+5)2a^2 + 45
= 4a^4 + 28a^2 + 45

(vii) (xyz – 4) (xyz – 2)

Answer: = x^2y^2z^2 + (-4 -2)xyz - 8
= x^2y^2z^2 - 6xyz - 8

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

(i) (b – 7)^2

Answer: = b^2 - 14b + 49

(ii) (xy + 3z)^2

Answer: = x^2y^2 + 6xyz + 9z^2

(iii) (6x^2– 5y)^2

Answer: = 36x^4 - 60x^2y + 25y^2

(iv) (2/3m+3/2n)^2

Answer: 4/9m^2+9/4n^2+2mn

(v) (0.4p – 0.5q)^2

Answer: = 0.16p^2 - 0.4pq + 0.25q^2

(vi) (2xy + 5y)

Answer: = 4x^2y^2 + 20xy^2 + 25y^2

Question 4: Simplify.

(i) (a^2–b^2)^2

Answer: = a^4-4a^2b^2+b^4

(ii) (2x + 5)^2– (2x – 5)^2

Answer: = 4x^2 + 20x +25 - (4x^2- 20x + 25)
= 4x^2 + 20x + 25 - 4x^2 + 20x - 25= 40x

(iii) (7m – 8n)^2+ (7m + 8n)^2

Answer: = 49m^2 - 112mn + 64n^2 + 49m^2 + 112mn + 64n^2
= 98m^2 + 128n^2

(iv) (4m + 5n)^2+ (5m + 4n)^2

Answer: = 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

Answer: = 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^2c

Answer: = a^2b^2 + 2ab^2c + b^2c^2 - 2ab^2c
= a^2b^2 + b^2c^2

(vii) (m^2– n^2m)^2+ 2m^3n^2

Answer: = m^4 - 2m^3n^2 + m^2n^4 + 2m^3n^2
= m^4 + m^2n^4