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# 7. Using $$a^2 – b^2 = (a + b) (a – b)$$, find (i) $$51^2 – 49^2$$ (ii) $$(1.02)^2 – (0.98)^2$$ (iii) $$153^2 – 147^2$$ (iv) $$12.1^2 – 7.9^2$$

(i) $$51^2 – 49^2 = (51 + 49) (51 – 49)$$

$$= 100 \times 2 = 200$$

(ii)$$(1.02)^2 – (0.98)^2 = (1.02 + 0.98) (1.02 – 0.98)$$

$$= 2.00 \times 0.04 = 0.08$$

(iii) $$153^2 – 147^2 = (153 + 147) (153 – 147)$$

$$= 300 \times 6 = 1800$$

(iv) $$12.1^2 – 7.9^2 = (12.1 + 7.9) (12.1 – 7.9)$$

$$= 20.0 \times 4.2 = 84$$

8. Using $$(x + a) (x + b) = x^2 + (a + b)x + ab$$, find)

(i) $$103 × 104$$

(ii) $$5.1 × 5.2$$

(iii) $$103 × 98$$

(iv) $$9.7 × 9.8$$

(i) $$103 \times 104 = (100 + 3)(100 + 4)$$

$$= (100)^2 + (3 + 4) (100) + 3 \times 4 = 10000 + 700 + 12$$

$$= 10712$$

(ii) $$5.1 \times 5.2 = (5 + 0.1) (5 + 0.2)$$

$$= (5)^2 + (0.1 + 0.2) (5) + 0.1 \times 0.2$$

$$= 25 + 1.5 + 0.02 = 26.5 + 0.02 = 26.52$$

(iii) $$103 \times 98 = (100 + 3) (100 – 2)$$

$$= (100)^2 + (3 – 2) (100) + 3 \times (-2) = 10000 + 100 – 6$$

$$= 10100 – 6 = 10094$$

(iv) $$9.7 \times 9.8 = (10 – 0.3) (10 – 0.2)$$

$$= (10)^2 – (0.3 + 0.2) (10) + (-0.3) (-0.2) = 100 – 5 + 0.06$$

$$= 95 + 0.06 = 95.06$$