Algebraic Expression

Exercise 9.5 Part 2

Question 5: Show that.

(i) `(3x + 7)^2– 84x = (3x – 7)^2`

Answer: LHS `= 9x^2 + 42x + 49 - 84x`
`= 9x^2- 42x + 49`
RHS `= 9x^2 - 42x + 49`
`LHS = RHS`

(ii) `(9p – 5q)^2+ 180pq = (9p + 5q)^2`

Answer: LHS `= 81p^2 - 90pq + 25q^2+ 180pq`
`= 81p^2 + 90pq + 25q^2`
RHS `= 81p^2 + 90pq + 25q^2`

(iii) `(4/3m-3/4n)^2+2mn=(16)/(9)m^2+(9)/(16)n^2`

Answer: `(16)/(9)m^2+(9)/(16)n^2-2mn+2mn`

`=(16)/(9)m^2+(9)/(16)n^2` = RHS proved

(iv) `(4pq + 3q)^2– (4pq – 3q)^2= 48pq^2`

Answer: `= 16p^2q^2 + 24pq^2 + 9q^2 - 16p^2q^2 + 24pq^2- 9q^2`
`= 48pq^2`

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

Answer:`= a^2 - b^2 + b^2 - c^2 + c^2 - a^2= 0`


Question 6: Using identities, evaluate.

(i) `71^2`

Answer: `71^2 = (70+1)^2`
Using `(a + b)^2 = a^2 + 2ab + b^2`
`= 70^2 + 140 + 1^2`
`= 4900 + 140 +1= 5041`

(ii) `99^2`

Answer: `= (100 -1)^2`
`= 100^2 - 200 + 1^2`
`= 10000 - 200 + 1= 9801`

(iii) `102^2`

Answer:`= (100 + 2)^2`
`= 100^2 + 400 + 2^2`
`= 10000 + 400 + 4= 10404`

(iv) `998^2`

Answer: `= (1000 - 2)^2`
`= 1000^2 - 4000 + 2^2`
`= 1000000 - 4000 + 4= 996004`

(v) `5.2^2`

Answer: `= (5 + 0.2)^2`
`= 5^2 + 2 + 0.2^2`
`= 25 + 2 + 0.4= 27.4`

(vi) `297 xx 303`

Answer: `= (300 - 3 )(300 + 3)`
Using `(a - b)(a + b) = a^2 - b^2`
`= 300^2 - 3^2`
`= 90000 - 9= 89991`

(vii) `78 xx 82`

Answer: `= (80 - 2)(80 + 2)`
`= 80^2 - 2^2`
`= 6400 - 4= 6396`

(viii) `8.9^2`

Answer: `= (9 - 0.1)^2`
`= 9^2 - 1.8 + 0.1^2`
`= 81 - 1.8 + 0.01= 79.21`

(ix) `10.5 xx 9.5`

Answer: `= (10 + 0.5)(10 - 0.5)`
`= 10^2 - 0.5^2`
`= 100 - 0.25= 99.75`


Question 7: Using `a^2– b^2 = (a + b) (a – b)`, find

(i) `51^2– 49^2`

Answer: `= (51 + 49)(51 - 49)`
`= 100 xx 2= 200`

(ii) `(1.02)^2– (0.98)^2`

Answer:`= (1.02 + 0.98)(1.02 - 0.98)`
`= 2 xx 0.04= 0.08`

(iii) `153^2– 147^2`

Answer: `= (153 + 147)(153 - 147)`
= 300 xx 6= 1800`

(iv) `12.1^2– 7.9^2`

Answer: `= (12.1 + 7.9)(12.1 - 7.9)`
`= 20 xx 4.2= 84`

Question 8: Using `(x + a) (x + b) = x^2+ (a + b) x + ab`, find

(i) `103 xx 104`

Answer:`= (100 + 3)(100 + 4)`
`= 100^2 + (3 + 4)100 + 12`
`= 10000 + 1200 + 12= 11212`

(ii) `5.1 xx 5.2`

Answer: `= (5 + 0.1)(5 + 0.2)`
`= 5^2 + (0.1+0.2)5 + 0.02`
`= 25 + 1.5 + 0.02= 26.52`

(iii) `103 xx 98`

Answer `= (100 + 3)(100 - 2)`
`= 100^2 + (3-2)100 - 6`
`= 10000 + 100 - 6= 10094`

(iv) `9.7 xx 9.8`

Answer: `= (9 + 0.7 )(9 + 0.8)`
`= 9^2 + (0.7 + 0.8)9 + 0.63`
`= 81 + 13.5 + 0.63= 95.13`



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