Algebraic Expressions

Exercise 9.5

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^2- 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 - 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= 40`

(iii) `(7m – 8n)^2+ (7m + 8n)^2`

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



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