Cho biểu thức \(P = \left( {\frac{{2\sqrt {xy} }}{{x - y}} - \frac{{\sqrt x + \sqrt y }}{{2\sqrt x - 2\sqrt y }}} \right).\frac{{2\sqrt x }}{{\sqrt x - \sqrt y }}\) (x, y ≥ 0, x ≠ y). Tính giá trị của P, biết \(\frac{x}{y} = \frac{4}{9}\). (Kết quả viết dưới dạng số thập phân)

Hướng dẫn giải

Đáp án: −0,4

Ta có: \(P = \left( {\frac{{2\sqrt {xy} }}{{x - y}} - \frac{{\sqrt x + \sqrt y }}{{2\sqrt x - 2\sqrt y }}} \right).\frac{{2\sqrt x }}{{\sqrt x - \sqrt y }}\)

             \( = \left[ {\frac{{4\sqrt {xy} }}{{2\left( {\sqrt x - \sqrt y } \right)\left( {\sqrt x + \sqrt y } \right)}} - \frac{{\sqrt x + \sqrt y }}{{2\left( {\sqrt x - \sqrt y } \right)}}} \right].\frac{{2\sqrt x }}{{\sqrt x - \sqrt y }}\)

             \( = \left[ {\frac{{4\sqrt {xy} }}{{2\left( {\sqrt x - \sqrt y } \right)\left( {\sqrt x + \sqrt y } \right)}} - \frac{{\left( {\sqrt x + \sqrt y } \right)\left( {\sqrt x + \sqrt y } \right)}}{{2\left( {\sqrt x - \sqrt y } \right)\left( {\sqrt x + \sqrt y } \right)}}} \right].\frac{{2\sqrt x }}{{\sqrt x - \sqrt y }}\)

            \( = \frac{{4\sqrt {xy} - {{\left( {\sqrt x + \sqrt y } \right)}^2}}}{{2\left( {\sqrt x - \sqrt y } \right)\left( {\sqrt x + \sqrt y } \right)}}.\frac{{2\sqrt x }}{{\sqrt x - \sqrt y }}\)

            \( = \frac{{4\sqrt {xy} - x - 2\sqrt {xy} - y}}{{\left( {\sqrt x - \sqrt y } \right)\left( {\sqrt x + \sqrt y } \right)}}.\frac{{2\sqrt x }}{{\sqrt x - \sqrt y }}\)

            \( = \frac{{ - \left( {x + 2\sqrt {xy} + y} \right)}}{{2\left( {\sqrt x - \sqrt y } \right)\left( {\sqrt x + \sqrt y } \right)}}.\frac{{2\sqrt x }}{{\sqrt x - \sqrt y }}\)

            \( = \frac{{ - {{\left( {\sqrt x + \sqrt y } \right)}^2}}}{{2\left( {\sqrt x - \sqrt y } \right)\left( {\sqrt x + \sqrt y } \right)}}.\frac{{2\sqrt x }}{{\sqrt x - \sqrt y }}\)

             \( = \frac{{ - \sqrt x }}{{\sqrt x + \sqrt y }}\)

Với \(\frac{x}{y} = \frac{4}{9}\) suy ra y = \(\frac{{9x}}{4}\).

Do đó, P \( = \frac{{ - \sqrt x }}{{\sqrt x + \sqrt y }} = \frac{{ - \sqrt x }}{{\sqrt x + \frac{3}{2}\sqrt x }} = \frac{{ - \sqrt x }}{{\frac{5}{2}\sqrt x }} = - \frac{2}{5} = - 0,4\).