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Dễ thây \(\hept{\begin{cases}\sqrt{a^2+2017}-a\ne0\\\sqrt{b^2+2017}-b\ne0\end{cases}}\)
\(\Rightarrow\hept{\begin{cases}\left(a+\sqrt{a^2+2017}\right)\left(\sqrt{a^2+2017}-a\right)\left(b+\sqrt{b^2+2017}\right)=2017\left(\sqrt{a^2+2017}-a\right)\\\left(a+\sqrt{a^2+2017}\right)\left(b+\sqrt{b^2+2017}\right)\left(\sqrt{b^2+2017}-b\right)=2017\left(\sqrt{b^2+2017}-b\right)\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}2017\left(b+\sqrt{b^2+2017}\right)=2017\left(\sqrt{a^2+2017}-a\right)\\2017\left(a+\sqrt{a^2+2017}\right)=2017\left(\sqrt{b^2+2017}-b\right)\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}b+\sqrt{b^2+2017}=\sqrt{a^2+2017}-a\\a+\sqrt{a^2+2017}=\sqrt{b^2+2017}-b\end{cases}}\)
\(\Leftrightarrow a+b=0\)
a) điều kiện xác định : \(a>2;a\ne11\)
ta có : \(P=\left(\dfrac{\sqrt{a-2}+2}{3}\right)\left(\dfrac{\sqrt{a-2}}{3+\sqrt{a-2}}+\dfrac{a+7}{11-a}\right):\left(\dfrac{3\sqrt{a-2}+1}{a-3\sqrt{a-2}-2}-\dfrac{1}{\sqrt{a-2}}\right)\)
\(\Leftrightarrow P=\left(\dfrac{\sqrt{a-2}+2}{3}\right)\left(\dfrac{\sqrt{a-2}}{3+\sqrt{a-2}}+\dfrac{a+7}{\left(3+\sqrt{a-2}\right)\left(3-\sqrt{a-2}\right)}\right):\left(\dfrac{3\sqrt{a-2}+1}{\sqrt{a-2}\left(\sqrt{a-2}-3\right)}-\dfrac{1}{\sqrt{a-2}}\right)\) \(\Leftrightarrow P=\left(\dfrac{\sqrt{a-2}+2}{3}\right)\left(\dfrac{\sqrt{a-2}\left(3-\sqrt{a-2}\right)+a+7}{\left(3+\sqrt{a-2}\right)\left(3-\sqrt{a-2}\right)}\right):\left(\dfrac{3\sqrt{a-2}+1-\sqrt{a-2}+3}{\sqrt{a-2}\left(\sqrt{a-2}-3\right)}\right)\) \(\Leftrightarrow P=\left(\dfrac{\sqrt{a-2}+2}{3}\right)\left(\dfrac{3\left(\sqrt{a-2}+3\right)}{\left(3+\sqrt{a-2}\right)\left(3-\sqrt{a-2}\right)}\right):\left(\dfrac{2\sqrt{a-2}+4}{\sqrt{a-2}\left(\sqrt{a-2}-3\right)}\right)\) \(\Leftrightarrow P=\left(\dfrac{\sqrt{a-2}+2}{3}\right)\left(\dfrac{3}{\left(3-\sqrt{a-2}\right)}\right)\left(\dfrac{\sqrt{a-2}\left(\sqrt{a-2}-3\right)}{2\left(\sqrt{a-2}+2\right)}\right)\) \(\Leftrightarrow P=\dfrac{-\sqrt{a-2}}{2}\)
ta có : \(a+b=\sqrt{2017-a^2}+\sqrt{2017-b^2}\)
\(\Leftrightarrow\left(a+b\right)\left(\sqrt{2017-a^2}-\sqrt{2017-b^2}\right)=b^2-a^2\)
\(\Leftrightarrow b-a=\sqrt{2017-a^2}-\sqrt{2017-b^2}\)
\(\Leftrightarrow2b=2\sqrt{2017-a^2}\Leftrightarrow b^2=2017-a^2\Rightarrow\left(đpcm\right)\)
Ta có : \(\left(x+\sqrt{x^2+2017}\right)\left(-x+\sqrt{x^2+2017}\right)=2017\left(1\right)\)
\(\left(y+\sqrt{y^2+2017}\right)\left(-y+\sqrt{y^2+2017}\right)=2017\left(2\right)\)
nhân theo vế của ( 1 ) ; ( 2 ) , ta có :
\(2017\left(-x+\sqrt{x^2+2017}\right)\left(-y+\sqrt{y^2+2017}\right)=2017^2\)
\(\Rightarrow\left(-x+\sqrt{x^2+2017}\right)\left(-y+\sqrt{y^2+2017}\right)=2017\)
rồi bạn nhân ra , kết hợp với việc nhân biểu thức ở phần trên xong cộng từng vế , cuối cùng ta đc :
\(xy+\sqrt{\left(x^2+2017\right)\left(y^2+2017\right)}=2017\)
\(\Leftrightarrow\sqrt{\left(x^2+2017\right)\left(y^2+2017\right)}=2017-xy\)
\(\Leftrightarrow x^2y^2+2017\left(x^2+y^2\right)+2017^2=2017^2-2\cdot2017xy+x^2y^2\)
\(\Rightarrow x^2+y^2=-2xy\Rightarrow\left(x+y\right)^2=0\Rightarrow x=-y\)
A = 2017
( phần trên mk lười nên không nhân ra, bạn giúp mk nhân ra nha :) )
2/ \(\frac{\sqrt{x-2011}-1}{x-2011}+\frac{\sqrt{y-2012}-1}{y-2012}+\frac{\sqrt{z-2013}-1}{z-2013}=\frac{3}{4}\)
\(\Leftrightarrow\frac{4\sqrt{x-2011}-4}{x-2011}+\frac{4\sqrt{y-2012}-4}{y-2012}+\frac{4\sqrt{z-2013}-4}{z-2013}=3\)
\(\Leftrightarrow\left(1-\frac{4\sqrt{x-2011}-4}{x-2011}\right)+\left(1-\frac{4\sqrt{y-2012}-4}{y-2012}\right)+\left(1-\frac{4\sqrt{z-2013}-4}{z-2013}\right)=0\)
\(\Leftrightarrow\left(\frac{x-2011-4\sqrt{x-2011}+4}{x-2011}\right)+\left(\frac{y-2012-4\sqrt{y-2012}+4}{y-2012}\right)+\left(\frac{z-2013-4\sqrt{z-2013}+4}{z-2013}\right)=0\)
\(\Leftrightarrow\frac{\left(\sqrt{x-2011}-2\right)^2}{x-2011}+\frac{\left(\sqrt{y-2012}-2\right)^2}{y-2012}+\frac{\left(\sqrt{z-2013}-2\right)^2}{z-2013}=0\)
Dấu = xảy ra khi \(\sqrt{x-2011}=2;\sqrt{y-2012}=2;\sqrt{z-2013}=2\)
\(\Leftrightarrow x=2015;y=2016;z=2017\)
Bài 1 bạn nhân \(\left(b-\sqrt{b^2+2017}\right)\)sau đó nó tạo thành hăng đẳng thức,sau đó tiếp tục nhân liên hợp,là ra a=-b
\(\Rightarrow a+b=0\)
1/ Ta có:
\(\hept{\begin{cases}\left(a+\sqrt{a^2+2017}\right)\left(\sqrt{a^2+2017}-a\right)\left(b+\sqrt{b^2+2017}\right)=2017\left(\sqrt{a^2+2017}-a\right)\\\left(a+\sqrt{a^2+2017}\right)\left(\sqrt{b^2+2017}-b\right)\left(b+\sqrt{b^2+2017}\right)=2017\left(\sqrt{b^2+2017}-b\right)\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}2017\left(b+\sqrt{b^2+2017}\right)=2017\left(\sqrt{a^2+2017}-a\right)\\2017\left(a+\sqrt{a^2+2017}\right)=2017\left(\sqrt{b^2+2017}-b\right)\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}b+\sqrt{b^2+2017}=\sqrt{a^2+2017}-a\left(1\right)\\a+\sqrt{a^2+2017}=\sqrt{b^2+2017}-b\left(2\right)\end{cases}}\)
Lấy (1) + (2) vế theo vế ta được
\(a+b=0\)