solve(256+(1/4x)^2=(1/2x)^2) Wie ist das ohne Taschenrechner zu lösen?

Guten Morgen Anonymous,

$${\mathtt{256}}{\mathtt{\,\small\textbf+\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{4}}}}{\mathtt{\,\times\,}}{\mathtt{x}}\right){\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{4}}}}{\mathtt{\,\times\,}}{\mathtt{x}}\right) = \left({\frac{{\mathtt{1}}}{{\mathtt{2}}}}{\mathtt{\,\times\,}}{\mathtt{x}}\right){\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{2}}}}{\mathtt{\,\times\,}}{\mathtt{x}}\right)$$

$${\mathtt{256}}{\mathtt{\,\small\textbf+\,}}{\frac{{{\mathtt{x}}}^{{\mathtt{2}}}}{{\mathtt{16}}}} = {\frac{{{\mathtt{x}}}^{{\mathtt{2}}}}{{\mathtt{4}}}}$$                          $${\frac{{{\mathtt{x}}}^{{\mathtt{2}}}}{{\mathtt{4}}}} = {\frac{{\mathtt{4}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}}{{\mathtt{16}}}}$$

$${\mathtt{256}} = {\frac{{\mathtt{4}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}}{{\mathtt{16}}}}{\mathtt{\,-\,}}{\frac{{\mathtt{1}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}}{{\mathtt{16}}}}$$

$${\mathtt{256}} = {\frac{{\mathtt{3}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{3}}}}{{\mathtt{16}}}}$$          =>       $${{\mathtt{x}}}^{{\mathtt{2}}} = {\frac{{\mathtt{256}}{\mathtt{\,\times\,}}{\mathtt{16}}}{{\mathtt{3}}}}$$          ;   $${\mathtt{256}}{\mathtt{\,\times\,}}{\mathtt{16}} = {{\mathtt{2}}}^{{\mathtt{8}}}{\mathtt{\,\times\,}}{{\mathtt{2}}}^{{\mathtt{4}}} = {{\mathtt{2}}}^{{\mathtt{12}}}$$

$${\mathtt{x}} = {\sqrt{{\frac{{{\mathtt{2}}}^{{\mathtt{12}}}}{{\mathtt{3}}}}}} = {{\mathtt{2}}}^{{\mathtt{6}}}{\mathtt{\,\times\,}}{\sqrt{{\frac{{\mathtt{1}}}{{\mathtt{3}}}}}}$$       =  $${\mathtt{64}}{\mathtt{\,\times\,}}{\sqrt{{\frac{{\mathtt{1}}}{{\mathtt{3}}}}}}$$           und    $${\mathtt{x}} = {\mathtt{\,-\,}}\left({\mathtt{64}}{\mathtt{\,\times\,}}{\sqrt{{\frac{{\mathtt{1}}}{{\mathtt{3}}}}}}\right)$$

Andere Schreibweise:     $${\mathtt{x}} = {\frac{{\mathtt{64}}}{{\sqrt{{\mathtt{3}}}}}}$$       und      $${\mathtt{x}} = {\mathtt{\,-\,}}{\frac{{\mathtt{64}}}{{\sqrt{{\mathtt{3}}}}}}$$

Gruß radix !