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July 2022 1 5 Report

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[tex] \frac{1}{ x^{2} } + \frac{x-2}{x} [/tex]

[tex] \frac{2mn}{ m^{2} + n^{2} } + \frac{2m}{ m^{2} - n^{2} } - \frac{1}{m-n} [/tex]

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Еpifanа $\frac{1}{ x^{2} } + \frac{x-2}{x} =\frac{1+x(x-2)}{ x^{2} }=\frac{1+x^{2} -2x}{ x^{2} }=\frac{(x-1)^2}{ x^{2} }$
$\frac{2mn}{ m^{2} + n^{2} } + \frac{2m}{ m^{2} - n^{2} } - \frac{1}{m-n} =\frac{2mn}{ m^{2} + n^{2} } + \frac{2m}{ (m - n)(m+n) } - \frac{1}{m-n}$ = $\frac{2mn}{ m^{2} + n^{2} } + \frac{2m-(m+n)}{ (m - n)(m+n) }=\frac{2mn}{ m^{2} + n^{2} } + \frac{2m-m-n}{ (m - n)(m+n) }$ = $\frac{2mn}{ m^{2} + n^{2} } + \frac{m-n}{ (m - n)(m+n) }=\frac{2mn}{ m^{2} + n^{2} } + \frac{1}{ m+n }$ = $\frac{2mn(m+n)+m^2+n^2}{(m^{2} + n^{2})(m+n)}=\frac{2m^2n+2mn^2+m^2+n^2}{(m^{2} + n^{2})(m+n)}$ = $\frac{2m^2n+2mn^2+m^2+n^2+m^2n-m^2n+mn^2-mn^2}{(m^{2} + n^{2})(m+n)}$ = $\frac{(3m^2n+3mn^2+m^2+n^2)-m^2n-mn^2}{(m^{2} + n^{2})(m+n)} = \frac{(m+n)^3-mn(m+n)}{(m^{2} + n^{2})(m+n)}$ = $\frac{(m+n)((m+n)^2-mn)}{(m^{2} + n^{2})(m+n)} = \frac{(m+n)^2-mn}{m^{2} + n^{2}}$ = $\frac{m^2+2mn+n^2-mn}{m^{2} + n^{2}} = \frac{m^2+mn+n^2}{m^{2} + n^{2}}$

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