Cho P= (x √x -3)/ (x- 2√x -3) – 2(√x -3 )/ (√x +1) = (√x +3) /( 3-√x)
a) Rút gọn P
b) Tính P khi x = 14 -6√5
c) Tính GTNN của P
( Giúp mình với ạ)
Cho P= (x √x -3)/ (x- 2√x -3) – 2(√x -3 )/ (√x +1) = (√x +3) /( 3-√x)
a) Rút gọn P
b) Tính P khi x = 14 -6√5
c) Tính GTNN của P
( Giúp mình với ạ)
Giải thích các bước giải:
Ta có;
\(\begin{array}{l}
a,\\
P = \dfrac{{x\sqrt x – 3}}{{x – 2\sqrt x – 3}} – \dfrac{{2.\left( {\sqrt x – 3} \right)}}{{\sqrt x + 1}} + \dfrac{{\sqrt x + 3}}{{3 – \sqrt x }}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( \begin{array}{l}
x \ge 0\\
x \ne 9
\end{array} \right)\\
= \dfrac{{x\sqrt x – 3}}{{\left( {\sqrt x – 3} \right)\left( {\sqrt x + 1} \right)}} – \dfrac{{2\left( {\sqrt x – 3} \right)}}{{\sqrt x + 1}} – \dfrac{{\sqrt x + 3}}{{\sqrt x – 3}}\\
= \dfrac{{\left( {x\sqrt x – 3} \right) – 2.{{\left( {\sqrt x – 3} \right)}^2} – \left( {\sqrt x + 3} \right)\left( {\sqrt x + 1} \right)}}{{\left( {\sqrt x – 3} \right)\left( {\sqrt x + 1} \right)}}\\
= \dfrac{{\left( {x\sqrt x – 3} \right) – 2.\left( {x – 6\sqrt x + 9} \right) – \left( {x + 4\sqrt x + 3} \right)}}{{\left( {\sqrt x – 3} \right)\left( {\sqrt x + 1} \right)}}\\
= \dfrac{{x\sqrt x – 3 – 2x + 12\sqrt x – 18 – x – 4\sqrt x – 3}}{{\left( {\sqrt x – 3} \right)\left( {\sqrt x + 1} \right)}}\\
= \dfrac{{x\sqrt x – 3x + 8\sqrt x – 24}}{{\left( {\sqrt x – 3} \right)\left( {\sqrt x + 1} \right)}}\\
= \dfrac{{x\left( {\sqrt x – 3} \right) + 8\left( {\sqrt x – 3} \right)}}{{\left( {\sqrt x – 3} \right)\left( {\sqrt x + 1} \right)}}\\
= \dfrac{{\left( {\sqrt x – 3} \right)\left( {x + 8} \right)}}{{\left( {\sqrt x – 3} \right)\left( {\sqrt x + 1} \right)}}\\
= \dfrac{{x + 8}}{{\sqrt x + 1}}\\
b,\\
x = 14 – 6\sqrt 5 = 9 – 2.3.\sqrt 5 + 5 = {\left( {3 – \sqrt 5 } \right)^2}\\
\Rightarrow \sqrt x = 3 – \sqrt 5 \\
P = \dfrac{{x + 8}}{{\sqrt x + 1}} = \dfrac{{14 – 6\sqrt 5 + 8}}{{3 – \sqrt 5 + 1}} = \dfrac{{22 – 6\sqrt 5 }}{{4 – \sqrt 5 }} = \dfrac{{\left( {22 – 6\sqrt 5 } \right)\left( {4 + \sqrt 5 } \right)}}{{\left( {4 – \sqrt 5 } \right)\left( {4 + \sqrt 5 } \right)}} = \dfrac{{58 – 2\sqrt 5 }}{{11}}\\
c,\\
P = \dfrac{{x + 8}}{{\sqrt x + 1}} = \dfrac{{\left( {x – 1} \right) + 9}}{{\sqrt x + 1}} = \dfrac{{\left( {\sqrt x – 1} \right)\left( {\sqrt x + 1} \right) + 9}}{{\sqrt x + 1}}\\
= \left( {\sqrt x – 1} \right) + \dfrac{9}{{\sqrt x + 1}} = \left[ {\left( {\sqrt x + 1} \right) + \dfrac{9}{{\sqrt x + 1}}} \right] – 2\\
\ge 2.\sqrt {\left( {\sqrt x + 1} \right).\dfrac{9}{{\sqrt x + 1}}} – 2 = 2.3 – 2 = 4\\
\Rightarrow {P_{\min }} = 4 \Leftrightarrow \sqrt x + 1 = \dfrac{9}{{\sqrt x + 1}} \Leftrightarrow \sqrt x + 1 = 3 \Leftrightarrow x = 4
\end{array}\)