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18 tháng 5 2021

`a)P=((1-asqrta)/(1-sqrta)+sqrta).((1+asqrta)/(1+sqrta)-sqrta)`

`=(((1-sqrta)(a+sqrta+1))/(1-sqrta)+sqrta).(((1+sqrta)(a-sqrta+1))/(1+sqrta)-sqrta)`

`=(a+sqrta+1+sqrta)(a-sqrta+1-sqrta)`

`=(a+2sqrta+1)(a-2sqrta+1)`

`=(sqrta+1)^2(sqrta-1)^2`

`=(a-1)^2`

`b)a<7-4sqrt3`

`<=>(a-1)^2<(2-sqrt3)^2`

`<=>sqrt3-2<a-1<2-sqrt3`

`<=>sqrt3-1<a<3-sqrt3`

AH
Akai Haruma
Giáo viên
2 tháng 3 2021

Đề bài có vẻ bị lỗi. Bạn xem lại đề. 

a) \(P=\dfrac{1-2\sqrt{a}+a}{1-\sqrt{a}}\cdot\dfrac{1+2\sqrt{a}+a}{1+\sqrt{a}}\) \(=\left(1-\sqrt{a}\right)\left(1+\sqrt{a}\right)\) \(=1-a\)

b) ĐKXĐ: \(\left\{{}\begin{matrix}a\ge0\\a\ne1\end{matrix}\right.\)

Để P>0 \(\Leftrightarrow1-a>0\) \(\Leftrightarrow a< 1\)

  Vậy \(0\le a< 1\)

30 tháng 1 2022

a) ĐKXĐ: \(a\ge0;a\ne1\)

\(P=\left(\dfrac{1-a\sqrt{a}}{1-\sqrt{a}}+\sqrt{a}\right)\left(\dfrac{1+a\sqrt{a}}{1+\sqrt{a}}-\sqrt{a}\right)\)

\(=\left[\dfrac{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}+a\right)}{1-\sqrt{a}}+\sqrt{a}\right]\left[\dfrac{\left(1+\sqrt{a}\right)\left(1-\sqrt{a}+a\right)}{1+\sqrt{a}}\right]\)

\(=\left(1+2\sqrt{a}+a\right)\left(1-2\sqrt{a}+a\right)\)

\(=\left(1-a\right)^2\)

b) Để \(P< 7-4\sqrt{3}\)

\(\Rightarrow\left(1-a\right)^2< 7-4\sqrt{3}\)

\(\Leftrightarrow\left|1-a\right|< \left(2-\sqrt{3}\right)^2\)

\(\Leftrightarrow\sqrt{3}-2< a-1< 2-\sqrt{3}\)

\(\Leftrightarrow\sqrt{3}-1< a< 3-\sqrt{3}\)

Vậy \(\sqrt{3}-1< a< 3-\sqrt{3}\) thì \(P< 7-4\sqrt{3}\)

30 tháng 1 2022

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20 tháng 10 2023

a: ĐKXĐ: \(\left\{{}\begin{matrix}a>0\\a< >1\end{matrix}\right.\)

\(P=\dfrac{a\sqrt{a}-1}{a-\sqrt{a}}-\dfrac{a\sqrt{a}+1}{a+\sqrt{a}}+\left(\sqrt{a}-\dfrac{1}{\sqrt{a}}\right)\cdot\left(\dfrac{3\sqrt{a}}{\sqrt{a}-1}-\dfrac{\sqrt{a}+2}{\sqrt{a}+1}\right)\)

\(=\dfrac{\left(\sqrt{a}-1\right)\left(a+\sqrt{a}+1\right)}{\sqrt{a}\left(\sqrt{a}-1\right)}-\dfrac{\left(\sqrt{a}+1\right)\left(a-\sqrt{a}+1\right)}{\sqrt{a}\left(\sqrt{a}+1\right)}+\dfrac{a-1}{\sqrt{a}}\cdot\dfrac{3\sqrt{a}\left(\sqrt{a}+1\right)-\left(\sqrt{a}+2\right)\left(\sqrt{a}-1\right)}{a-1}\)

\(=\dfrac{a+\sqrt{a}+1-\left(a-\sqrt{a}+1\right)}{\sqrt{a}}+\dfrac{3a+3\sqrt{a}-a-\sqrt{a}+2}{\sqrt{a}}\)

\(=\dfrac{2\sqrt{a}+2a+2\sqrt{a}+2}{\sqrt{a}}=\dfrac{2\left(\sqrt{a}+1\right)^2}{\sqrt{a}}\)

b: \(P=\sqrt{a}+7\)

=>\(2\left(a+2\sqrt{a}+1\right)=a+7\sqrt{a}\)

=>\(2a+4\sqrt{a}+2-a-7\sqrt{a}=0\)

=>\(a-3\sqrt{a}+2=0\)

=>\(\left(\sqrt{a}-1\right)\left(\sqrt{a}-2\right)=0\)

=>\(\left[{}\begin{matrix}a=1\left(loại\right)\\a=4\left(nhận\right)\end{matrix}\right.\)

c: \(P-6=\dfrac{2\left(\sqrt{a}+1\right)^2-6\sqrt{a}}{\sqrt{a}}\)

\(=\dfrac{2a+4\sqrt{a}+2-6\sqrt{a}}{\sqrt{a}}=\dfrac{2a-2\sqrt{a}+2}{\sqrt{a}}\)

\(=\dfrac{2\left(a-\sqrt{a}+\dfrac{1}{4}+\dfrac{3}{4}\right)}{\sqrt{a}}=\dfrac{2\left[\left(\sqrt{a}-\dfrac{1}{2}\right)^2+\dfrac{3}{4}\right]}{\sqrt{a}}>0\)

=>P>6

5 tháng 7 2021

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a: \(P=\dfrac{a-\sqrt{a}}{\sqrt{a}\left(\sqrt{a}-1\right)}\cdot\dfrac{\sqrt{a}-1}{1}=\sqrt{a}-1\)

b: Để P<0 thì căn a-1<0

=>căn a<1

=>0<a<1

29 tháng 12 2022

\(P=\left(\dfrac{\sqrt{a}}{\sqrt{a}-1}-\dfrac{\sqrt{a}}{a-\sqrt{a}}\right):\dfrac{\sqrt{a}+1}{a-1}\left(a>0;a\ne1\right)\)

\(a,P=\left(\dfrac{\sqrt{a}}{\sqrt{a}-1}-\dfrac{\sqrt{a}}{a-\sqrt{a}}\right):\dfrac{\sqrt{a}+1}{a-1}\)

\(=\left[\dfrac{\sqrt{a}}{\sqrt{a}-1}-\dfrac{\sqrt{a}}{\sqrt{a}\left(\sqrt{a}-1\right)}\right]:\dfrac{\sqrt{a}+1}{a-1}\)

\(=\left(\dfrac{\sqrt{a}}{\sqrt{a}-1}-\dfrac{1}{\sqrt{a}-1}\right):\dfrac{\sqrt{a}+1}{a-1}\)

\(=1:\dfrac{\sqrt{a}+1}{a-1}\)

\(=\dfrac{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}{\sqrt{a}+1}\)

\(=\sqrt{a}-1\)

\(b,P< 0\Rightarrow\sqrt{a}-1< 0\Leftrightarrow\sqrt{a}< 1\Leftrightarrow a< 1\)

Kết hợp điều kiện \(a>0;a\ne1\)

\(\Rightarrow0< a< 1\)

Sửa đề: \(Q=\left(\dfrac{1}{\sqrt{a}-1}-\dfrac{1}{\sqrt{a}}\right):\left(\dfrac{\sqrt{a}+1}{\sqrt{a}-2}-\dfrac{\sqrt{a}-1}{\sqrt{a}+2}\right)\)

a) ĐKXĐ: \(\left\{{}\begin{matrix}a\ge0\\a\notin\left\{1;4\right\}\end{matrix}\right.\)

Ta có: \(Q=\left(\dfrac{1}{\sqrt{a}-1}-\dfrac{1}{\sqrt{a}}\right):\left(\dfrac{\sqrt{a}+1}{\sqrt{a}-2}-\dfrac{\sqrt{a}-1}{\sqrt{a}+2}\right)\)

\(=\dfrac{\sqrt{a}-\sqrt{a}+1}{\sqrt{a}\left(\sqrt{a}-1\right)}:\dfrac{a+3\sqrt{a}+2-a+3\sqrt{a}-2}{\left(\sqrt{a}-2\right)\left(\sqrt{a}+2\right)}\)

\(=\dfrac{1}{\sqrt{a}\left(\sqrt{a}-1\right)}\cdot\dfrac{\left(\sqrt{a}-2\right)\left(\sqrt{a}+2\right)}{6\sqrt{a}}\)

\(=\dfrac{a-4}{6a\left(\sqrt{a}-1\right)}\)

c) Thay \(a=9-4\sqrt{5}\) vào Q, ta được:

\(Q=\dfrac{5-4\sqrt{5}}{6\left(9-4\sqrt{5}\right)\left(\sqrt{5}-3\right)}\)

\(=\dfrac{5-4\sqrt{5}}{6\left(9\sqrt{5}-27-20+12\sqrt{5}\right)}\)

\(=\dfrac{5-4\sqrt{5}}{6\left(21\sqrt{5}-47\right)}\)

\(=\dfrac{\left(5-4\sqrt{5}\right)\left(21\sqrt{5}+47\right)}{-24}\)

\(=\dfrac{105\sqrt{5}+235-420-188\sqrt{5}}{-24}\)

\(=\dfrac{-83\sqrt{5}-185}{-24}=\dfrac{83\sqrt{5}+185}{24}\)

10 tháng 7 2021

cảm ơn ạ!