3. Using some algebraic identity , evaluate the following 
(a) (1007)²
(b) (93)²
(c) 996*1004
(d) 503*495

Dear Student,

$$\begin{gathered} \left(a\right) {\left(1007\right)}^2 = {\left(1000+7\right)}^2 \\ \mathit{U}\mathit{s}\mathit{i}\mathit{n}\mathit{g}\mathbf{ }\mathit{I}\mathit{d}\mathit{e}\mathit{n}\mathit{t}\mathit{i}\mathit{t}\mathit{y}: {\left(a+b\right)}^2 = a^2+b^2+2ab \\ = {\left(1000\right)}^2+{\left(7\right)}^2+2\times 1000\times 7 \\ = 1000000+49+14000 \\ = 1014049 \\ \left(b\right) {\left(93\right)}^2 = {\left(100-7\right)}^2 \\ \mathit{U}\mathit{s}\mathit{i}\mathit{n}\mathit{g}\mathbf{ }\mathit{I}\mathit{d}\mathit{e}\mathit{n}\mathit{t}\mathit{i}\mathit{t}\mathit{y}: {\left(a-b\right)}^2 = a^2+b^2-2ab \\ = {\left(100\right)}^2+{\left(7\right)}^2-2\times 100\times 7 \\ = 10000+49-1400 \\ = 8649 \\ \left(c\right) 996\times 1004 \\ = \left(1000-4\right)\left(1000+4\right) \\ \mathit{U}\mathit{s}\mathit{i}\mathit{n}\mathit{g}\mathbf{ }\mathit{I}\mathit{d}\mathit{e}\mathit{n}\mathit{t}\mathit{i}\mathit{t}\mathit{y} \left(x+y\right)\left(x-y\right) = x^2-y^2 \\ where x= 1000, y =4 \\ = {\left(1000\right)}^2-{\left(4\right)}^2 \\ = 1000000-16 \\ = 999984 \\ \left(d\right) 503\times 495 \\ \mathit{U}\mathit{s}\mathit{i}\mathit{n}\mathit{g}\mathbf{ }\mathit{I}\mathit{d}\mathit{e}\mathit{n}\mathit{t}\mathit{i}\mathit{t}\mathit{y}\mathbf{:}\mathbf{ }\left(x+a\right)\left(x+b\right)\mathbf{ }= x^2+\left(a+b\right)x+ab \\ where x = 500, a= 3, b=-5 \\ = {\left(500\right)}^2+\left(3-5\right)500+3\times -5 \\ = 250000+\left(-2\right)500-15 \\ = 250000-1000-15 \\ = 248985 \end{gathered}$$
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