Simplify each of the following products:
$ (\frac{x}{2}-\frac{2}{5})(\frac{2}{5}-\frac{x}{2})-x^{2}+2 x $

Given:

\( (\frac{x}{2}-\frac{2}{5})(\frac{2}{5}-\frac{x}{2})-x^{2}+2 x \)

To do:

We have to simplify the given product.

Solution:

We know that,

$(a+b)^2=a^2+b^2+2ab$

$(a-b)^2=a^2+b^2-2ab$

$(a+b)(a-b)=a^2-b^2$

Therefore,

$(\frac{x}{2}-\frac{2}{5})(\frac{2}{5}-\frac{x}{2})-x^{2}+2 x=(\frac{x}{2}-\frac{2}{5})(\frac{-x}{2}+\frac{2}{5})-x^{2}+2 x$

$=-(\frac{x}{2}-\frac{2}{5})(\frac{x}{2}-\frac{2}{5})-x^{2}+2 x$

$=-(\frac{x}{2}-\frac{2}{5})^{2}-x^{2}+2 x$

$=-[(\frac{x}{2})^{2}+(\frac{2}{5})^{2}-2 \times \frac{x}{2} \times \frac{2}{5}]-x^{2}+2 x$

$=-[\frac{x^{2}}{4}+\frac{4}{25}-\frac{2 x}{5}]-x^{2}+2 x$

$=\frac{-x^{2}}{4}-\frac{4}{25}+\frac{2 x}{5}-x^{2}+2 x$

$=\frac{-x^{2}}{4}-x^{2}+\frac{2 x}{5}+2 x-\frac{4}{25}$

$=-(\frac{x^{2}+4 x^{2}}{4})+\frac{2 x+10 x}{5}-\frac{4}{25}$

$=-\frac{5 x^{2}}{4}+\frac{12 x}{5}-\frac{4}{25}$

Hence, $(\frac{x}{2}-\frac{2}{5})(\frac{2}{5}-\frac{x}{2})-x^{2}+2 x=-\frac{5 x^{2}}{4}+\frac{12 x}{5}-\frac{4}{25}$.

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Updated on: 10-Oct-2022

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