Toán Lớp 9: BT1: a)Rút gọn:
A= (x-y/căn x – căn y + x căn x – y căn y/y-x) : ( căn x – căn y)^2 + căn xy/ căn x + căn y
b) Chứng minh A > hoặc bằng 0
BT2:Rút gọn: B= (a^2 + căn a/ a – căn a +1) – (2a+ căn a/ căn a) +1
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Toán Lớp 9: BT1: a)Rút gọn:
A= (x-y/căn x – căn y + x căn x – y căn y/y-x) : ( căn x – căn y)^2 + căn xy/ căn x + căn y
b) Chứng minh A > hoặc bằng 0
BT2:Rút gọn: B= (a^2 + căn a/ a – căn a +1) – (2a+ căn a/ căn a) +1
Comments ( 1 )
a)Dkxd:x,y \ge 0;x\# y\\
A = \left( {\dfrac{{x – y}}{{\sqrt x – \sqrt y }} + \dfrac{{x\sqrt x – y\sqrt y }}{{y – x}}} \right):{\left( {\sqrt x – \sqrt y } \right)^2}\\
+ \dfrac{{\sqrt {xy} }}{{\sqrt x + \sqrt y }}\\
= \left( {\sqrt x + \sqrt y – \dfrac{{\left( {\sqrt x – \sqrt y } \right)\left( {x + \sqrt {xy} + y} \right)}}{{\left( {\sqrt x – \sqrt y } \right)\left( {\sqrt x + \sqrt y } \right)}}} \right)\\
:{\left( {\sqrt x – \sqrt y } \right)^2} + \dfrac{{\sqrt {xy} }}{{\sqrt x + \sqrt y }}\\
= \left( {\sqrt x + \sqrt y – \dfrac{{x + \sqrt {xy} + y}}{{\sqrt x + \sqrt y }}} \right):{\left( {\sqrt x – \sqrt y } \right)^2} + \dfrac{{\sqrt {xy} }}{{\sqrt x + \sqrt y }}\\
= \dfrac{{x + 2\sqrt {xy} + y – x – \sqrt {xy} – y}}{{\sqrt x + \sqrt y }}.\dfrac{1}{{{{\left( {\sqrt x – \sqrt y } \right)}^2}}} + \dfrac{{\sqrt {xy} }}{{\sqrt x + \sqrt y }}\\
= \dfrac{{\sqrt {xy} }}{{\sqrt x + y}}.\dfrac{1}{{{{\left( {\sqrt x – \sqrt y } \right)}^2}}} + \dfrac{{\sqrt {xy} }}{{\sqrt x + \sqrt y }}\\
= \left( {\dfrac{1}{{{{\left( {\sqrt x – \sqrt y } \right)}^2}}} + 1} \right).\dfrac{{\sqrt {xy} }}{{\sqrt x + \sqrt y }} \ge 0\\
Do:x;y \ge 0\\
Vay\,A \ge 0\\
Bt2)\\
B = \left( {\dfrac{{{a^2} + \sqrt a }}{{a – \sqrt a + 1}}} \right) – \dfrac{{2a + \sqrt a }}{{\sqrt a }} – 1\\
= \dfrac{{\sqrt a \left( {a\sqrt a + 1} \right)}}{{a – \sqrt a + 1}} – \dfrac{{\sqrt a \left( {2\sqrt a + 1} \right)}}{{\sqrt a }} – 1\\
= \dfrac{{\sqrt a \left( {\sqrt a + 1} \right)\left( {a – \sqrt a + 1} \right)}}{{a – \sqrt a + 1}} – \left( {2\sqrt a + 1} \right) – 1\\
= \sqrt a \left( {\sqrt a + 1} \right) – 2\sqrt a – 1 + 1\\
= a + \sqrt a – 2\sqrt a \\
= a – \sqrt a
\end{array}$