Question

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$$\begin{gathered} 11.\mathrm{If}\mathrm{x}+\mathrm{y}=8\mathrm{and}\mathrm{xy}=3\frac{3}{4},\mathrm{find}\mathrm{the}\mathrm{values}\mathrm{of} \\ \left(\mathrm{i}\right)\mathrm{x}-\mathrm{y}\left(\mathrm{ii}\right)3\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)\left(\mathrm{iii}\right)5\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)+4\left(\mathrm{x}-\mathrm{y}\right). \end{gathered}$$

Answer 1

Dear student

$$\begin{gathered} \mathrm{Given}:\mathrm{x}+\mathrm{y}=8 \\ \mathrm{xy}=3\frac{3}{4}=\frac{15}{4} \\ 1)\mathrm{Consider}:{\left(\mathrm{x}-\mathrm{y}\right)}^2 \\ ={\left(\mathrm{x}+\mathrm{y}\right)}^2-4\mathrm{xy} \\ ={\left(8\right)}^2-4\left(\frac{15}{4}\right) \\ =64-15 \\ =49 \\ \mathrm{So},\mathrm{x}-\mathrm{y}=\sqrt{49}=\pm 7 \\ 2)3\left({\mathrm{x}}^2+{\mathrm{y}}^2\right) \\ =3\left[{\left(\mathrm{x}+\mathrm{y}\right)}^2-2\mathrm{xy}\right]{}^{{}^{}}\left[\mathrm{as}{\left(\mathrm{a}+\mathrm{b}\right)}^2={\mathrm{a}}^2+{\mathrm{b}}^2+2\mathrm{ab}\right] \\ =3\left[8^2-2\left(\frac{15}{4}\right)\right] \\ =3\left[64-\frac{15}{2}\right] \\ =3\left[\frac{128-15}{2}\right] \\ =3\left[\frac{113}{2}\right] \\ =\frac{339}{2} \\ 3)5\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)+4\left(\mathrm{x}-\mathrm{y}\right) \\ \mathrm{Case}\mathrm{I}\mathrm{when}\mathrm{x}-\mathrm{y}=7 \\ =5\left(\frac{113}{2}\right)+4\times 7 \\ =\frac{565}{2}+28 \\ =\frac{621}{2} \\ \mathrm{Case}\mathrm{II}\mathrm{when}\mathrm{x}-\mathrm{y}=-7 \\ 5\left(\frac{113}{2}\right)+4\times \left(-7\right) \\ =\frac{565}{2}-28 \\ =\frac{509}{2} \end{gathered}$$
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