8th Maths 9.5

Q1.`text{What is an Identity ?}`
`text{Answer}`
`text{An equality, true for every value of the variable }``text{in it, is called an identity.}`
1. `text{(a + b)}^2 = text{a}^2 + 2text{ab} + text{b}^2`
2. `text{(a - b)}^2 = text{a}^2 - 2text{ab} + text{b}^2`
3. `text{(a + b)(a - b) = a}^2 - text{b}^2`
4. `text{(x + a)(x + b) = x}^2 text{+ (a + b)x + ab}`

Exercise 9.5

Q1. 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) `(1.1m - 0.4)(1.1m + 0.4)`
vi) `(a^2 + b^2)(- a^2 + b^2)`
vii) `(6x - 7)(6x + 7)`
viii) `(- a + c)(- a + c)`
ix) `(x/2 + (3y)/4)(x/2 + (3y)/4)`
x) `(7a - 9b)(7a - 9b)`


i) `(x + 3)(x + 3)`
`text{Sol.}`
`=(x+3)(x+3)`
`=(x+3)^2`
`[text{ (a + b)}^2 = text{a}^2 + 2text{ab} + text{b}^2 ]`
`text{where a = x, b = 3`
`= x^2 + 2 timesx times 3 + (3)^2`
`= x^2 + 6x + 9`
`text{Answer}`
`=x^2 + 6x + 9`

ii) `(2y + 5)(2y + 5)`
`text{Sol.}`
`=(2y + 5)(2y + 5)`
`=(2y+5)^2`
`[text{ (a + b)}^2 = text{a}^2 + 2text{ab} + text{b}^2 ]`
`text{where a = }2y, text{b = 5}`
`= (2y)^2 + 2 times 2y times 5 + (5)^2`
`= 4y^2 + 20y + 25`
`text{Answer}`
`= 4y^2 + 20y + 25`

iii) `(2a - 7)(2a - 7)`
`text{Sol.}`
`=(2a - 7)(2a - 7)`
`=(2a - 7)^2`
`[text{ (a - b)}^2 = text{a}^2 - 2text{ab} + text{b}^2 ]`
`text{where a }= 2a, text{b }= 7`
`= (2a)^2 + 2 times 2a times 7 + (7)^2`
`= 4a^2 + 28a + 49`
`text{Answer}`
`= 4a^2 + 28a + 49`

iv) `(3a - 1/2)(3a - 1/2)`
`text{Sol.}`
`=(3a - 1/2)(3a - 1/2)`
`=(3a - 1/2)^2`
`[text{ (a - b)}^2 = text{a}^2 - 2text{ab} + text{b}^2 ]`
`text{where a } = 3a, text{b } = 1/2`
`=(3a)^2 - 2 times 3a times 1/2 + (1/2)^2`
`= 9a^2 - 3a + 1/4`
`text{Answer}`
`= 9a^2 - 3a + 1/4`

v) `(1.1m - 0.4)(1.1m + 0.4)`
`text{Sol.}`
`=(1.1m - 0.4)(1.1m + 0.4)`
`[text{ (a + b)(a - b) = a}^2 - text{b}^2]`
`text{Where a = 1.1m, b = 0.4}`
`= (1.1m)^2-(0.4)^2`
`=1.21m^2 - 1.6`
`text{Answer}`
`=1.21m^2 - 1.6`

vi) `(a^2 + b^2)(- a^2 + b^2)`
`text{Sol.}`
`=(a^2 + b^2)(-a^2 + b^2)`
`=(b^2+a^2)(b^2-a^2)`
`[text{ (a + b)(a - b) = a}^2 - text{b}^2]`
`text{Where }a = b^2, b = a^2`
`= (b^2)^2-(a^2)^2`
`=b^4 - a^4`
`text{Answer}`
`=b^4 - a^4`

vii) `(6x - 7)(6x + 7)`
`text{Sol.}`
`=(6x - 7)(6x + 7)`
`[text{ (a + b)(a - b) = a}^2 - text{b}^2]`
`text{Where a = 6x, b = 7}`
`= (6x)^2-(7)^2`
`=36x^2 - 49`
`text{Answer}`
`=36x^2 - 49`

viii) `(- a + c)(- a + c)`
`text{Sol.}`
`=(- a + c)(- a + c)`
`=(c - a)(c - a )`
`=(c - a)^2`
`[text{ (a - b)}^2 = text{a}^2 - 2text{ab} + text{b}^2 ]`
`text{where a = c, b = }a`
`= c^2 - 2 times c times a + a^2`
`= c^2 - 2ac + a^2`
`text{Answer}`
`= a^2 + 2ac +c^2`

ix) `(x/2 + (3y)/4)(x/2 + (3y)/4)`
`text{Sol.}`
`=(x/2 + (3y)/4)(x/2 + (3y)/4)`
`=(x/2 + (3y)/4)^2`
`[text{ (a + b)}^2 = text{a}^2 + 2text{ab} + text{b}^2 ]`
`text{where a} = x/2, text{b} = (3y)/4`
`= (x/2)^2 + 2 times x/2 times (3y)/4 + ((3y)/4)^2`
`= x^2/4 + (3xy)/4 + (9y^2)/16`
`text{Answer}`
`= x^2/4 + (3xy)/4 + (9y^2)/16`

x) `(7a - 9b)(7a - 9b)`
`text{Sol.}`
`=(7a - 9b)(7a - 9b)`
`=(7a - 9b)^2`
`[text{ (a - b)}^2 = text{a}^2 - 2text{ab} + text{b}^2 ]`
`text{where a = 7a, b = 9b}`
`= (7a)^2 - 2 times7a times 9b + (9b)^2`
`= 49a^2 - 126ab + 81b^2`
`text{Answer}`
`= 49a^2 - 126ab + 81b^2`

Q2. Use the identity `(x + a)(x + b) = x^2 + (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 + 3y)`
vi) `(2a^2 + 9)(2a^2 + 5)`
vii) `(xyz - 4)(xyz - 2)`

i) `(x + 3)(x + 7)`
`text{Sol.}`
`=(x + 3)(x + 7)`
`[text{ (x + a)(x + b) = x}^2 text{+ (a + b)x + ab }]`
`text{where x = x, a = 3, b = 7}`
`=x^2+(3+7)x+3times7`
`=x^2+10x+21`
`text{Answer}`
`=x^2+10x+21`

ii) `(4x + 5)(4x + 1)`
`text{Sol.}`
 `=(4x + 5)(4x + 1)`
`[text{ (x + a)(x + b) = x}^2 text{+ (a + b)x + ab }]`
`text{where x = 4x, a = 5, b = 1}`
`=(4x)^2+(5+1)4x+5times1`
`=16x^2+24x+5`
`text{Answer}`
`=16x^2+24x+5`

iii) `(4x - 5)(4x - 1)`
`text{Sol.}`
`=(4x - 5)(4x - 1)`
`[text{ (x - a)(x - b) = x}^2 text{- (a + b)x + ab }]`
`text{where x = 4x, a = 5, b = 1}`
`=(4x)^2-(5+1)4x+5times1`
`=16x^2-24x+5`
`text{Answer}`
`=16x^2-24x+5`

iv) `(4x + 5)(4x - 1)`
`text{Sol.}`
`=(4x + 5)(4x - 1)`
`[text{ (x + a)(x - b) = x}^2 text{+ (a - b)x - ab }]`
`text{where x = 4x, a = 5, b = 1}`
`=(4x)^2+(5-1)4x-5times1`
`=16x^2+16x-5`
`text{Answer}`
`=16x^2+16x-5`

v) `(2x + 5y)(2x + 3y)`
`text{Sol.}`
`=(2x + 5y)(2x + 3y)`
`[text{ (x + a)(x + b) = x}^2 text{+ (a + b)x + ab }]`
`text{where x = 2x, a = 5y, b = 3y}`
`=(2x)^2+(5y+3y)2x+5ytimes3y`
`=4x^2+16xy+15y^2`
`text{Answer}`
`=4x^2+16xy+15y^2`

vi) `(2a^2 + 9)(2a^2 + 5)`
`text{Sol.}`
`=(2a^2 + 9)(2a^2 + 5)`
`[text{ (x + a)(x + b) = x}^2 text{+ (a + b)x + ab }]`
`text{where x} = 2a^2, text{a } = 9, text{b } = 5`
`=(2a^2)^2+(9+5)2a^2+9times5`
`=4a^4+28a^2+45`
`text{Answer}`
`=4a^4+28a^2+45`

vii) `(xyz - 4)(xyz - 2)`
`text{Sol.}`
`=(xyz - 4)(xyz - 2)`
`[text{ (x - a)(x - b) = x}^2 text{- (a + b)x + ab }]`
`text{where x = xyz, a = 4, b = 2}`
`=(xyz)^2-(4+2)xyz+4times2`
`=x^2y^2z^2-6xyz+8`
`text{Answer}`
`=x^2y^2z^2-6xyz+8`


Q3. Find the following squares by using the identities.
i) `(b - 7)^2`
ii) `(xy - 3z)^2`
iii) `(6x^2 - 5y)^2`
iv) `(2/3m + 3/2n)^2`
v) `(0.4p - 0.5q)^2`
vi) `(2xy + 5y)^2`

i) `(b - 7)^2`
`text{Sol.}`
`=(b - 7)^2`
`[text{ (a - b)}^2 = text{a}^2 - 2text{ab} + text{b}^2 ]`
`text{where a }= b, text{b }= 7`
`= (b)^2 - 2 times b times 7 + (7)^2`
`= b^2 - 14b + 49`
`text{Answer}`
`= b^2 - 14b + 49`

ii) `(xy - 3z)^2`
`text{Sol.}`
`=(xy - 3z)^2`
`[text{ (a - b)}^2 = text{a}^2 - 2text{ab} + text{b}^2 ]`
`text{where a = xy, b = 3z}`
`= (xy)^2 - 2 times xy times 3z + (3z)^2`
`= x^2y^2 - 6xyz + 9z^2`
`text{Answer}`
`= x^2y^2 - 6xyz + 9z^2`

iii) `(6x^2 - 5y)^2`
`text{Sol.}`
`=(6x^2 - 5y)^2`
`[text{ (a - b)}^2 = text{a}^2 - 2text{ab} + text{b}^2 ]`
`text{where a} = 6x^2, text{b } = 5y`
`= (6x^2)^2 - 2 times 6x^2 times 5y + (5y)^2`
`= 36x^4 - 60x^2y + 25y^2`
`text{Answer}`
`= 36x^4 - 60x^2y + 25y^2`

iv) `(2/3m + 3/2n)^2`
`text{Sol.}`
`=(2/3m + 3/2n)^2`
`[text{ (a + b)}^2 = text{a}^2 + 2text{ab} + text{b}^2 ]`
`text{where a = }2/3m, text{b = }3/2n`
`= (2/3m)^2 + 2 times 2/3m times 3/2n + (3/2n)^2`
`= 4/9m^2 + 2mn + 9/4n^2`
`text{Answer}`
`= 4/9m^2 + 2mn + 9/4n^2`

v) `(0.4p - 0.5q)^2`
`text{Sol.}`
`=(0.4p - 0.5q)^2`
`[text{ (a - b)}^2 = text{a}^2 - 2text{ab} + text{b}^2 ]`
`text{where a = 0.4p, b = 0.5q}`
`= (0.4p)^2 - 2 times 0.4p times 0.5q + (0.5q)^2`
`= 0.16p^2 - 0.04pq + 0.25q^2`
`text{Answer}`
`= 0.16p^2 - 0.04pq + 0.25q^2`

vi) `(2xy + 5y)^2`
`text{Sol.}`
`=(2xy + 5y)^2`
`[text{ (a + b)}^2 = text{a}^2 + 2text{ab} + text{b}^2 ]`
`text{where a = }2xy, text{b = 5y}`
`= (2xy)^2 + 2 times 2xy times 5y + (5y)^2`
`= 4x^2y^2 + 20xy + 25y^2`
`text{Answer}`
`= 4x^2y^2 + 20xy + 25y^2`

Q4. Simplify.
i) `(a^2 - b^2)^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 - 2ab^2c`
vii) `(m^2 - n^2m)^2 + 2m^3n^2`

i) `(a^2 - b^2)^2`
`text{Sol.}`
`=(a^2 - b^2)^2`
`[text{ (a - b)}^2 = text{a}^2 - 2text{ab} + text{b}^2 ]`
`text{where a } = a^2, text{b } = b^2`
`= (a^2)^2 - 2 times a^2 times b^2+ (b^2)^2`
`= a^4 - 2a^2b^2 + b^4`
`text{Answer}`
`= a^4 - 2a^2b^2 + b^4`

ii) `(2x + 5)^2 - (2x - 5)^2`
`text{Sol.}`
`=(2x + 5)^2 - (2x - 5)^2`
`[text{  a}^2-text{b}^2=text{(a+b)(a-b)  }]`
`text{Where a = (2x+5), b = (2x-5)}`
`=(2x+5+2x-5)(2x+5-2x+5)`
`=4x times 10`
`=40x`
`text{Answer}`
`=40x`

iii) `(7m - 8n)^2 + (7m + 8n)^2`
`text{Sol.}`
`=(7m - 8n)^2 + (7m + 8n)^2`
`=(7m)^2-2times7m times 8n+(8n)^2``+(7m)^2+2times7m times 8n+(8n)^2`
`=49m^2-112mn+64n^2``+49m^2+112mn+64n^2`
`=98m^2+128n^2`
`text{Answer}`
`=98m^2+128n^2`

iv) `(4m + 5n)^2 + (5m + 4n)^2`
`text{Sol.}`
`=(4m + 5n)^2 + (5m + 4n)^2`
`=(4m)^2+2times4m times 5n+(5n)^2``+(5m)^2+2times5m times 4n+(4n)^2`
`=16m^2+40mn+25n^2``+25m^2+40mn+16n^2`
`=41m^2+80mn+41n^2`
`text{Answer}`
`=41m^2+80mn+41n^2`

v) `(2.5p -1.5q)^2 - (1.5p - 2.5q)^2`
`text{Sol.}`
`=(2.5p -1.5q)^2 - (1.5p - 2.5q)^2`
`=(2.5p)^2-2times2.5p times 1.5q+(1.5q)^2``- (1.5p)^2-2times1.5p times 2.5q+(2.5q)^2`
`=6.25p^2-7.50pq+2.25q^2``-2.25p^2+7.50pq-6.25q^2`
`=4p^2-4q^2`
`text{Answer}`
`=4p^2-4q^2`

vi) `(ab + bc)^2 - 2ab^2c`
`text{Sol.}`
`=(ab + bc)^2 - 2ab^2c`
`=(ab)^2+2timesabtimesbc+(bc)^2-2ab^2c`
`=a^2b^2+2ab^2c+b^2c^2-2ab^2c`
`=a^2b^2+b^2c^2`
`text{Answer}`
`=a^2b^2+b^2c^2`

vii) `(m^2 - n^2m)^2 + 2m^3n^2`
`text{Sol.}`
 `=(m^2 - n^2m)^2 + 2m^3n^2`
`=(m^2)^2-2timesm^2timesn^2m+(n^2m)^2+2m^3n^2`
`=m^4-2m^3n^2+n^4m^2+2m^3n^2`
`=m^4+n^4m^2`
`text{Answer}`
`=m^4+n^4m^2`

Q5. Show that.
i) `(3x + 7)^2 - 84x = (3x - 7)^2`
ii) `(9p - 5q)^2 + 180pq = (9p + 5q)^2`
iii) `(4/3m - 3/4n)^2 + 2mn = 16/9m^2 + 9/16n^2`
iv) `(4pq + 3q)^2 - (4pq - 3q)^2 = 48pq^2`
v) `(a - b)(a + b) + (b - c)(b + c) + (c - a)(c + a)`` = 0`

i) `(3x + 7)^2 - 84x = (3x - 7)^2`
`text{Sol.}`
`=text{LHS.}`
`=(3x + 7)^2 - 84x`
`=(3x)^2+2times3xtimes7+(7)^2-84x`
`=9x^2+42x+49-84x`
`=9x^2-42x+49`
`=(3x)^2-2times3xtimes7+(7)^2`
`=(3x-7)^2`
`=text{RHS.}`
`text{LHS.=RHS.}`
`text{Proved}`

ii) `(9p - 5q)^2 + 180pq = (9p + 5q)^2`
`text{Sol.}`
`text{LHS.}`
`=(9p - 5q)^2 + 180pq`
`=(9p)^2-2times9ptimes5q+(5q)^2+180pq`
`=81p^2-90pq+25q^2+180pq`
`=81p^2+90pq+25q^2`
`=(9p)^2+2times9ptimes5q+(5q)^2`
`=(9p+5q)^2`
`=text{RHS.}`
`text{LHS.=RHS.}`
`text{Proved}`

iii) `(4/3m - 3/4n)^2 + 2mn = 16/9m^2 + 9/16n^2`
`text{Sol.}`
`text{LHS.}`
`=(4/3m - 3/4n)^2 + 2mn`
`=(4/3m)^2-2times4/3m times3/4n+(3/4n)^2+2mn`
`=16/9m^2-2mn+9/16n^2+2mn`
`=16/9m^2+9/16n^2`
`text{RHS.}`
`text{LHS.=RHS.}`
`text{Proved}`

iv) `(4pq + 3q)^2 - (4pq - 3q)^2 = 48pq^2`
`text{Sol.}`
`=(4pq + 3q)^2 - (4pq - 3q)^2`
`=(4pq+3q+4pq-3q)(4pq+3q-4pq+3q)`
`=(8pq)(6q)`
`=48pq^2`
`text{LHS.}`
`text{LHS.=RHS.}`
`text{Proved}`

v) `(a - b)(a + b) + (b - c)(b + c) + (c - a)(c + a) = 0`
`text{Sol.}`
`text{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`
`text{RHS.}`
`text{LHS.=RHS.}`
`text{Proved}`

Q6. Using identities, evaluate
i) `71^2`
ii) `99^2`
iii) `102^2`
iv) `998^2`
v) `5.2^2`
vi) `297times303`
vii) `78times82`
viii) `8.9^2`
ix) `1.05times9.5`

i) `71^2`
`text{Sol.}`
`=71^2`
`=(70+1)^2`
`[text{ (a + b)}^2 = text{a}^2 + 2text{ab} + text{b}^2 ]`
`text{where a = }70, text{b = 1}`
`=(70)^2+2times70times1+(1)^2`
`=4900+140+1`
`=5041`
`text{Answer}`
`=5041`

ii) `99^2`
`text{Sol.}`
`=99^2`
`=(100-1)^2`
`[text{ (a + b)}^2 = text{a}^2 - 2text{ab} + text{b}^2 ]`
`text{where a = }100, text{b = 1}`
`=(100)^2-2times100times1+(1)^2`
`=10000-200+1`
`=9801`
`text{Answer}`
`=9801`

iii) `102^2`
`text{Sol.}`
`=102^2`
`=(100+2)^2`
`[text{ (a + b)}^2 = text{a}^2 + 2text{ab} + text{b}^2 ]`
`text{where a = }100, text{b = 2}`
`=(100)^2+2times100times2+(2)^2`
`=10000+400+4`
`=10404`
`text{Answer}`
`=10404`

iv) `998^2`
`text{Sol.}`
`=998^2`
`=(1000-2)^2`
`[text{ (a + b)}^2 = text{a}^2 - 2text{ab} + text{b}^2 ]`
`text{where a = }1000, text{b = 2}`
`=(1000)^2-2times1000times2+(2)^2`
`=1000000-4000+4`
`=996004`
`text{Answer}`
`=996004`

v) `5.2^2`
`text{Sol.}`
`=5.2^2`
`=(5.0+0.2)^2`
`[text{ (a + b)}^2 = text{a}^2 + 2text{ab} + text{b}^2 ]`
`text{where a = }5.0, text{b = 0.2}`
`=(5.0)^2+2times5.0times.2+(0.2)^2`
`=25+2.0+0.04`
`=27.04`
`text{Answer}`
`=27.04`


vi) `297times303`
`text{Sol.}`
`=297times303`
`=(300-3)(300+3)`
`[text{ (a + b)(a - b) = a}^2 - text{b}^2]`
`text{Where a = 300, b = 3}`
`=(300)^2-(3)^2`
`=90000-9`
`=89991`
`text{Answer}`
`=89991`

vii) `78times82`
`text{Sol.}`
`=78times82`
`=(80-2)(80+2)`
`[text{ (a + b)(a - b) = a}^2 - text{b}^2]`
`text{Where a = 80, b = 2}`
`=(80)^2-(2)^2`
`=6400-4`
`=6396`
`text{Answer}`
`=6396`

viii) `8.9^2`
`text{Sol.}`
`=8.9^2`
`=(8+0.9)^2`
`[text{ (a + b)}^2 = text{a}^2 + 2text{ab} + text{b}^2 ]`
`text{where a = }8, text{b = 0.9}`
`=(8)^2+2times8times0.9+(0.9)^2`
`=64+14.4+0.81`
`=79.21`
`text{Answer}`
`=79.21`

ix) `1.05times9.5`
`text{Sol.}`
`=1.05times9.5`
`=(1+0.05)(9+0.5)`
`=1(9+0.5)+0.05(9+0.5)`
`=1times 9 + 1t times 0.5 + 0.05 times 9 + 0.05 times 0.5` 
`=9+0.5+0.45+0.025`
`=9.5+0.475`
`=9.975`
`text{Answer}`
`=9.975`


Q7. Using `a^2 - b^2 = (a + b)(a - b)`, find
i) `51^2 - 48^2`
ii) `(1.02)^2 - (0.98)^2`
iii) `153^2 - 147^2`
iv) `12.1^2 - 7.9^2`

i) `51^2 - 49^2`
`text{Sol.}`
`=51^2 - 49^2`
`=(51+49)(51-49)`
`=100times2`
`=200`
`text{Answer}`
`=200`

ii) `(1.02)^2 - (0.98)^2`
`text{Sol.}`
`=(1.02)^2 - (0.98)^2`
`=(1.02+0.98)(1.02-0.98)`
`=2times0.04`
`=0.08`
`text{Answer}`
`=0.08`

iii) `153^2 - 147^2`
`text{Sol.}`
`=153^2 - 147^2`
`=(153+147)(153-147)`
`=300times6`
`=1800`
`text{Answer}`
`=1800`

iv) `12.1^2 - 7.9^2`
`text{Sol.}`
`=12.1^2 - 7.9^2`
`=(12.1+7.9)(12.1-7.9)`
`=20times4.2`
`=84`
`text{Answer}`
`=84`

Q8. Using `(x + a)(x + b) = x^2 + (a + b)x + ab`, find
i) `103times 104`
ii) `5.1times5.2`
iii) `103times98`
iv) `9.7times9.8`

i) `103times 104`
`text{Sol.}`
`=103times 104`
`=(100+3)(100+4)`
`=(100)^2+(3+4)100+3times4`
`=10000+700+12`
`=10712`
`text{Answer}`
`=10712`

ii) `5.1times5.2`
`text{Sol.}`
`=5.1times5.2`
`=(5+0.1)(5+0.2)`
`=(5)^2+(0.1+0.2)5+0.1times0.2`
`=25+0.3times5+0.02`
`=25+1.5+0.02`
`=26.52`
`text{Answer}`
`=26.52`

iii) `103times98`
`text{Sol.}`
`=103times98`
`=(100+3)(100-2)`
`=(100)^2+(3-2)100-3times2`
`=10000+100-6`
`=10000+94`
`=10094`
`text{Answer}`
`=10094`

iv) `9.7times9.8`
`text{Sol.}`
`=9.7times9.8`
`=(9+0.7)(9+0.8)`
`=(9)^2+(0.7+0.8)9+0.7times0.8`
`=81+1.5times9+0.56`
`=81+13.5+0.56`
`=95.06`
`text{Answer}`
`=95.06`

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