[tex]m^{2} - 2mn + k^{2} - 2nk + 2n^{2} = 0 \\ \\ m^{2} - 2mn + n^{2} + k^{2} - 2nk + n^{2} = 0 \\ \\ (m-n)^{2} + (k - n)^{2} = 0 \\ \\ m-n = 0 \ \hbox{and} \ k-n = 0 \\ \\ m = n \ \hbox{and} \ k = n[/tex]
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[tex]2mn - k^{2} = 25 \\ \\ 2n^{2} - n^{2} = 25 \\ \\ n^{2} = 25 \\ \\ n = 5 \ \vee \ n= -5[/tex]
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[tex] \dfrac{(m+n)^{2}}{2k} = \dfrac{(n+n)^{2}}{2n} = \dfrac{(2n)^{2}}{2n} = \dfrac{4n^{2}}{2n} = 2n \\ \\ 2n = 10 \ \vee \ 2n = -10[/tex]
