If \({\rm{x}} = \frac{{\sqrt {{\rm{a\;}} + {\rm{\;}}3{\rm{b}}} {\rm{\;}} + {\rm{\;}}\sqrt {{\rm{a}} - 3{\rm{b}}} }}{{\sqrt {{\rm{a\;}} + {\rm{\;}}3{\rm{b}}} - \sqrt {{\rm{a}} - 3{\rm{b}}} }}\) , then find the value of \(3{\rm{b}}{{\rm{x}}^2} - {\rm{\;}}2{\rm{ax\;}} + {\rm{\;}}3{\rm{b}} .\)
If \({\rm{x}} = \frac{{\sqrt {{\rm{a\;}} + {\rm{\;}}3{\rm{b}}} {\rm{\;}} + {\rm{\;}}\sqrt {{\rm{a}} - 3{\rm{b}}} }}{{\sqrt {{\rm{a\;}} + {\rm{\;}}3{\rm{b}}} - \sqrt {{\rm{a}} - 3{\rm{b}}} }}\) , then find the value of \(3{\rm{b}}{{\rm{x}}^2} - {\rm{\;}}2{\rm{ax\;}} + {\rm{\;}}3{\rm{b}} .\)
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1 answers
Given $(\frac{{\rm{x}}}{1} = \frac{{\sqrt {{\rm{a\;}} + {\rm{\;}}3{\rm{b}}} {\rm{\;}} + {\rm{\;}}\sqrt {{\rm{a}} - 3{\rm{b}}} }}{{\sqrt {{\rm{a\;}} + {\rm{\;}}3{\rm{b}}} - \sqrt {{\rm{a}} - 3{\rm{b}}} }})$
$(\Rightarrow {\rm{\;}}\frac{{{\rm{x\;}} + {\rm{\;}}1}}{{{\rm{x}} - 1}} = \frac{{(\sqrt {{\rm{a\;}} + {\rm{\;}}3{\rm{b}}} {\rm{\;}} + {\rm{\;}}\sqrt {{\rm{a}} - 3{\rm{b}})} {\rm{\;}} + {\rm{\;}}(\sqrt {{\rm{a\;}} + {\rm{\;}}3{\rm{b}}} - \sqrt {{\rm{a}} - 3{\rm{b}})} }}{{(\sqrt {{\rm{a\;}} + {\rm{\;}}3{\rm{b}}} {\rm{\;}} + {\rm{\;}}\sqrt {{\rm{a}} - 3{\rm{b}})} - (\sqrt {{\rm{a\;}} + {\rm{\;}}3{\rm{b}}} - \sqrt {{\rm{a}} - 3{\rm{b}})} }})$ (By componendo and dividendo)
$(\Rightarrow {\rm{\;}}\frac{{{\rm{x\;}} + {\rm{\;}}1}}{{{\rm{x}} - 1}} = \frac{{2\sqrt {{\rm{a\;}} + {\rm{\;}}3{\rm{b}}} }}{{2\sqrt {{\rm{a}} - 3{\rm{b}})} }})$
$(\Rightarrow {\rm{\;}}\frac{{{{\left( {{\rm{x\;}} + {\rm{\;}}1} \right)}^2}}}{{{{\left( {{\rm{x}} - 1} \right)}^2}}} = \frac{{{\rm{a\;}} + {\rm{\;}}3{\rm{b}}}}{{{\rm{a}} - 3{\rm{b}}}})$ (On squaring both sides)
$(\Rightarrow {\rm{\;}}\frac{{{{\left( {{\rm{x\;}} + {\rm{\;}}1} \right)}^2}{\rm{\;}} + {\rm{\;}}{{\left( {{\rm{x}} - 1} \right)}^2}}}{{{{\left( {{\rm{x\;}} + {\rm{\;}}1} \right)}^2} - {{\left( {{\rm{x}} - 1} \right)}^2}}} = \frac{{\left( {{\rm{a\;}} + {\rm{\;}}3{\rm{b}}} \right){\rm{\;}} + {\rm{\;}}\left( {{\rm{a}} - 3{\rm{b}}} \right)}}{{\left( {{\rm{a\;}} + {\rm{\;}}3{\rm{b}}} \right) - \left( {{\rm{a}} - 3{\rm{b}}} \right)}})$ (By componendo and dividendo)
$(\Rightarrow {\rm{\;}}\frac{{2{{\rm{x}}^2}{\rm{\;}} + {\rm{\;}}2}}{{4{\rm{x}}}} = \frac{{2{\rm{a}}}}{{6{\rm{b}}}}{\rm{\;}})$
$(\Rightarrow {\rm{\;}}\frac{{{{\rm{x}}^2}{\rm{\;}} + {\rm{\;}}1}}{{2{\rm{x}}}} = \frac{{\rm{a}}}{{3{\rm{b}}}})$
$(\Rightarrow {\rm{\;}}3{\rm{b}}{{\rm{x}}^2}{\rm{\;}} + {\rm{\;}}3{\rm{b}} = 2{\rm{ax}})$
$(\Rightarrow {\rm{\;}}3{\rm{b}}{{\rm{x}}^2} - 2{\rm{ax\;}} + {\rm{\;}}3{\rm{b}} = 0.)$