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a,a=b+1
suy ra a-b=1 suy ra(\(\sqrt{a}+\sqrt{b}\))(\(\sqrt{a}-\sqrt{b}\))=1
suy ra \(\sqrt{a}-\sqrt{b}\)=\(\frac{1}{\sqrt{a}+\sqrt{b}}\)(1)
vì a=b+1 suy ra a>b suy ra \(\sqrt{a}>\sqrt{b}\)suy ra \(\sqrt{a}+\sqrt{b}>2\sqrt{b}\)
suy ra \(\frac{1}{\sqrt{a}+\sqrt{b}}< \frac{1}{2\sqrt{b}}\)(2)
từ (1) ,(2) suy ra\(\sqrt{a}-\sqrt{b}< \frac{1}{2\sqrt{b}}\)suy ra \(2\left(\sqrt{a}-\sqrt{b}\right)< \frac{1}{\sqrt{b}}\)(*)
ta lại có b+1=c+2 suy ra b-c =1 suy ra\(\left(\sqrt{b}-\sqrt{c}\right)\left(\sqrt{b}+\sqrt{c}\right)=1\)
suy ra \(\sqrt{b}-\sqrt{c}=\frac{1}{\sqrt{b}+\sqrt{c}}\)(3)
vì b>c suy ra \(\sqrt{b}>\sqrt{c}\) suy ra \(\sqrt{b}+\sqrt{c}>2\sqrt{c}\)
suy ra \(\frac{1}{\sqrt{b}+\sqrt{c}}< \frac{1}{2\sqrt{c}}\)(4)
Từ (3),(4) suy ra \(\sqrt{b}-\sqrt{c}< \frac{1}{2\sqrt{c}}\) suy ra\(2\left(\sqrt{b}+\sqrt{c}\right)< \frac{1}{\sqrt{c}}\)(**)
từ (*),(**) suy ra đccm
\(A=\frac{\left(2\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\sqrt{5}+2\right)\left(\sqrt{5}+1\right)-\left(10+2\sqrt{5}\right)\left(\sqrt{5}-1\right)}{5-1}-1\)
\(=\frac{10+2\sqrt{5}+2\sqrt{5}+2-10\sqrt{5}+10-10+2\sqrt{5}}{4}-1\)
\(=\frac{12-4\sqrt{5}}{4}-1\)
\(=\frac{4\left(3-\sqrt{5}\right)}{4}-1\)
\(=3-\sqrt{5}-1\)
\(=2-\sqrt{5}\)
(còn biểu thức B hình như sai đề, bạn coi lại đề)
đề câu B nè : \(B=\sqrt{\left(1-\sqrt{2014}\right)^2}.\sqrt{2015+2\sqrt{2014}}\)
Bài 2 : đã cm bên kia
Bài 1: :|
we had điều này:
\(2=\frac{2014}{x}+\frac{2014}{y}+\frac{2014}{z}\)
\(\Leftrightarrow\frac{x-2014}{x}+\frac{y-2014}{y}+\frac{z-204}{z}=1\)
Xòng! bunyakovsky
P/s : Bệnh lười kinh niên tái phát nên ít khi ol sorry :<
b,\(B=\sqrt{1+2014^2+\dfrac{2014^2}{2015^2}}+\dfrac{2014}{2015}\)
Ta có :\(\left(2014+1\right)^2=2014^2+1+2.2014\)
\(\Rightarrow2014^2+1=2015^2-2.2014\)
\(\Rightarrow B=\sqrt{2015^2-2.2014+\left(\dfrac{2014}{2015}\right)^2}+\dfrac{2014}{2015}\)
\(=\sqrt{\left(2015-\dfrac{2014}{2015}\right)^2}+\dfrac{2014}{2015}\)
\(=2015-\dfrac{2014}{2015}+\dfrac{2014}{2015}\)
\(=2015\)
Vậy B=2015
c/ ĐKXĐ: \(x\ge3\)
\(\Leftrightarrow\sqrt{\left(x-1\right)\left(x-2\right)}+\sqrt{x-3}-\sqrt{x-2}-\sqrt{\left(x-1\right)\left(x+3\right)}=0\)
\(\Leftrightarrow\left(\sqrt{\left(x-1\right)\left(x-2\right)}-\sqrt{x-2}\right)-\left(\sqrt{\left(x-1\right)\left(x+3\right)}-\sqrt{x+3}\right)=0\)
\(\Leftrightarrow\sqrt{x-2}\left(\sqrt{x-1}-1\right)-\sqrt{x+3}\left(\sqrt{x-1}-1\right)=0\)
\(\Leftrightarrow\left(\sqrt{x-2}-\sqrt{x+3}\right)\left(\sqrt{x-1}-1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x-2}-\sqrt{x+3}=0\\\sqrt{x-1}-1=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x-2}=\sqrt{x+3}\\\sqrt{x-1}=1\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x-2=x+3\left(vn\right)\\x=2< 3\left(ktm\right)\end{matrix}\right.\)
Vậy pt đã cho vô nghiệm
c,\(\left(\frac{\sqrt{1+a}}{\sqrt{1+a}-\sqrt{1-a}}+\frac{1-a}{\sqrt{1-a^2}-1+a}\right)\left(\sqrt{\frac{1}{a^2}-1}-\frac{1}{a}\right)\)
\(=\left(\frac{\sqrt{1+a}}{\sqrt{1+a}-\sqrt{1-a}}+\frac{\sqrt{1-a}.\sqrt{1-a}}{\sqrt{1-a}\left(\sqrt{1+a}-\sqrt{1-a}\right)}\right)\left(\frac{\sqrt{1-a^2}-1}{a}\right)\)
\(=\frac{\left(\sqrt{1+a}+\sqrt{1-a}\right)^2}{\left(1+a\right)-\left(1-a\right)}.\frac{\left(\sqrt{1-a^2}-1\right)}{a}=-1\)
M chỉ làm tiếp thôi nha, ko chép lại đề với đk đâu
a,
\(=\frac{a+2\sqrt{ab}+b-4\sqrt{ab}}{\sqrt{a}-\sqrt{b}}-\)\(\frac{\sqrt{ab}\left(\sqrt{a}-\sqrt{b}\right)}{\sqrt{ab}}\)
\(=\frac{a-2\sqrt{ab}+b}{\sqrt{a}-\sqrt{b}}-\left(\sqrt{a}-\sqrt{b}\right)\)
\(=\frac{\left(\sqrt{a}-\sqrt{b}\right)^2}{\sqrt{a}-\sqrt{b}}-\sqrt{a}+\sqrt{b}\)
\(=\sqrt{a}-\sqrt{b}-\sqrt{a}+\sqrt{b}\)
\(=0\)
b,
\(=\left(a-b\right)\left(\sqrt{\frac{a+b}{a-b}}-1\right)\left(a-b\right)\left(\sqrt{\frac{a+b}{a-b}}+1\right)\)
\(=\left(a-b\right)^2\left(\frac{a+b}{a-b}-1\right)\)
\(=\left(a-b\right)^2\cdot\frac{a+b-a+b}{a-b}\)
\(=\left(a-b\right)2b=2ab-2b^2\)
b, Ta có \(2015^2=\left(2014+1\right)^2=2014^2+2.2014+1\)
=> \(2014^2+1=2015^2-2.2014\)
=> \(B=\sqrt{1+2014^2+\frac{2014^2}{2015^2}}+\frac{2014}{2015}\)
= \(\sqrt{2015^2-2.2014+\frac{2014^2}{2015^2}}+\frac{2014}{2015}\)
= \(\sqrt{\left(2015-\frac{2014}{2015}\right)^2}+\frac{2014}{2015}\) = \(2015-\frac{2014}{2015}+\frac{2014}{2015}=2015\)
=> đpcm