\(\left\{{}\begin{matrix}\dfrac{7}{\sqrt{x}-7}-\dfrac{4}{\sqrt{y}+6}=\dfrac{5}{3}.\\\dfrac{5}{\sqrt{x}-7}+\dfrac{3}{\sqrt{y}+6}=2\dfrac{1}{6}.\end{matrix}\right.\) \(\left(x,y\ge0;x\ne49\right).\)

\(\Leftrightarrow\left\{{}\begin{matrix}7\dfrac{1}{\sqrt{x}-7}-4\dfrac{1}{\sqrt{y}+6}=\dfrac{5}{3}.\\5\dfrac{1}{\sqrt{x}-7}+3\dfrac{1}{\sqrt{y}+6}=\dfrac{13}{6}.\end{matrix}\right.\)

Đặt \(\dfrac{1}{\sqrt[]{x}-7}=a\); \(\dfrac{1}{\sqrt[]{y}+6}=b\left(a,b\ne0\right).\)

\(\Rightarrow\left\{{}\begin{matrix}7a-4b=\dfrac{5}{3}.\\5a+3b=\dfrac{13}{6}.\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}a=\dfrac{1}{3}.\\b=\dfrac{1}{6}.\end{matrix}\right.\) \(\left(TM\right).\)

\(\Rightarrow\left\{{}\begin{matrix}\dfrac{1}{\sqrt{x}-7}=\dfrac{1}{3}.\\\dfrac{1}{\sqrt{y}+6}=\dfrac{1}{6}.\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{x}-7=3.\\\sqrt{y}+6=6.\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{x}=10.\\\sqrt{y}=0.\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x=100\left(TM\right).\\y=0\left(TM\right).\end{matrix}\right.\)

Vậy hệ phương trình có nghiệm duy nhất là: \(\left(x;y\right)=\left(100;0\right).\)

có sai đề ko z bạn

Lời giải:
a)

Nhân $\sqrt{2}$ vào PT(1) và $\sqrt{3}$ vào PT(2) ta có:

HPT \(\Leftrightarrow \left\{\begin{matrix} \sqrt{6}x-4y=7\sqrt{2}\\ \sqrt{6}x+9y=-6\sqrt{2}\end{matrix}\right.\)

\(\Rightarrow (\sqrt{6}x-4y)-(\sqrt{6}x+9y)=13\sqrt{2}\)

\(\Leftrightarrow -13y=13\sqrt{2}\Rightarrow y=-\sqrt{2}\)

\(\Rightarrow x=\frac{7+2\sqrt{2}y}{\sqrt{3}}=\sqrt{3}\)

Vậy..............

b)

Nhân $2+\sqrt{3}$ vào PT(1) và $(\sqrt{2}+1)$ vào PT(2) thu được:

\(\left\{\begin{matrix} (\sqrt{2}+1)(2+\sqrt{3})x-y=2(2+\sqrt{3})\\ (2+\sqrt{3})(\sqrt{2}+1)+y=2(\sqrt{2}+1)\end{matrix}\right.\)

Trừ theo vế:

\(\Rightarrow -2y=2(2+\sqrt{3})-2(\sqrt{2}+1)=2+2\sqrt{3}-2\sqrt{2}\)

\(\Rightarrow y=\sqrt{2}-\sqrt{3}-1\)

\(\Rightarrow x=\frac{2+(2-\sqrt{3})y}{\sqrt{2}+1}=1+\sqrt{2}-\sqrt{3}\)

Vậy.........

\(\left\{{}\begin{matrix}\sqrt{x^2+3}+2\sqrt{x}=3+\sqrt{y}\left(1\right)\\\sqrt{y^2+3}+2\sqrt{y}=3+\sqrt{x}\left(2\right)\end{matrix}\right.\)\(\left(đk;x;y\ge0\right)\)

\(\left(1\right)-\left(2\right)\Rightarrow\sqrt{x^2+3}+2\sqrt{x}-\sqrt{y^2+3}-2\sqrt{y}=\sqrt{y}-\sqrt{x}\)

\(\Leftrightarrow\sqrt{x^2+3}-\sqrt{y^2+3}+2\sqrt{x}-2\sqrt{y}+\sqrt{x}-\sqrt{y}=0\left(3\right)\)

\(với:x=y=0\Rightarrow ko\) \(là\) \(nghiệm\)

\(vỡi:x=y\ne0\Rightarrow x;y>0\)

\(\left(3\right)\Leftrightarrow\dfrac{x^2+3-y^2-3}{\sqrt{x^2+3}+\sqrt{y^2+3}}+\dfrac{4x-4y}{2\sqrt{x}+2\sqrt{y}}+\dfrac{x-y}{\sqrt{x}+\sqrt{y}}=0\)

\(\Leftrightarrow\left(x-y\right)\left[\dfrac{x+y}{\sqrt{x^2+3}+\sqrt{y^2+3}}+\dfrac{4}{2\sqrt{x}+2\sqrt{y}}+\dfrac{1}{\sqrt{x}+\sqrt{y}}>0\left(\forall x;y>0\right)\right]=0\)

\(\Rightarrow x=y\left(4\right)\)

\(\left(4\right)và\left(1\right)\Rightarrow\sqrt{x^2+3}+2\sqrt{x}=3+\sqrt{x}\Leftrightarrow\sqrt{x^2+3}+\sqrt{x}-3=0\)

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

a) \(x^2-|x|-6=0\)(1)

Với \(x\ge0\)=> \(|x|=x\) 

Phương trình trở thành

\(x^2-x-6=0\)

\(\left(a=1,b=-1,c=-6\right)\)

\(\Delta=b^2-4ac=\left(-1\right)^2-4\cdot1\cdot\left(-6\right)=1+24=25>0\)

=>\(\sqrt{\Delta}=\sqrt{25}=5\)

=> Phương trình có 2 nghiệm

\(x_1=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-\left(-1\right)+5}{2\cdot1}=3\)(thỏa)

\(x_2=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-\left(-1\right)-5}{2\cdot1}=-2\)(loại)

Với \(x< 0\)=> \(|x|=-x\) 

Phương trình trở thành

\(-x^2+x-6=0\)

\(\left(a=-1,b=1,c=-6\right)\)

\(\Delta=b^2-4ac=1^2-4\cdot\left(-1\right)\cdot\left(-6\right)=1-24=-23< 0\)

=> Phương trình vô nghiệm

Vậy nghiệm của phuong trình (1) là x=3

Câu 1:

\(ĐK:x\ge2\)

Áp dụng BĐT cauchy ta có:

\(\left(x+1\right)+4\ge2\sqrt{4\left(x+1\right)}=4\sqrt{x+1}\\ \Leftrightarrow2\sqrt{x+1}\le\dfrac{x+5}{2}\)

Ta có \(\left(x-2\right)+1\ge2\sqrt{x-2}\Leftrightarrow\sqrt{x-2}\le\dfrac{x-1}{2}\)

\(\Leftrightarrow P\le\dfrac{x+5}{2}+\dfrac{x-1}{2}-x+2013=x+2-x+2013=2015\)

Dấu \("="\Leftrightarrow\left\{{}\begin{matrix}x+1=4\\x-2=1\end{matrix}\right.\Leftrightarrow x=3\)

Câu 2:

\(HPT\Leftrightarrow\left\{{}\begin{matrix}10\sqrt{x}+15y^3=140\\4y^3-10\sqrt{x}=12\end{matrix}\right.\left(x\ge0\right)\\ \Leftrightarrow19y^3=152\\ \Leftrightarrow y^3=8\Leftrightarrow y=2\\ \Leftrightarrow2\sqrt{x}+24=28\Leftrightarrow\sqrt{x}=2\Leftrightarrow x=4\)

Vậy \(\left(x;y\right)=\left(4;2\right)\)

Câu 3:

\(HPT\Leftrightarrow\left\{{}\begin{matrix}x=y+2\\my+2m+y=3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=y+2\\y=\dfrac{3-2m}{m+1}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{5}{m+1}\\x=\dfrac{3-2m}{m+1}\end{matrix}\right.\\ \Leftrightarrow xy=\dfrac{5\left(3-2m\right)}{\left(m+1\right)^2}\)

Đặt \(xy=t\)

\(\Leftrightarrow m^2t+2mt+t=15-10m\\ \Leftrightarrow m^2t+2m\left(t+5\right)+t-15=0\)

PT có nghiệm nên \(\Delta'=\left(t+5\right)^2-t\left(t-15\right)\ge0\)

\(\Leftrightarrow10t+25+15t\ge0\Leftrightarrow t\ge-1\)

Vậy \(xy_{min}=-1\Leftrightarrow\dfrac{5\left(2m-3\right)}{\left(m+1\right)^2}=1\Leftrightarrow m^2-8m+16=0\Leftrightarrow m=4\)

Câu 4: \(a^2+b^2=4a+bc+540\)

c đâu ra vậy?

Câu 5:

Thay \(x=3\Leftrightarrow P\left(2\right)+2P\left(2\right)=3^2\Leftrightarrow P\left(2\right)=3\)

Thay \(x=\sqrt{2013}\)

\(\Leftrightarrow P\left(\sqrt{2013}-1\right)+2P\left(2\right)=\left(\sqrt{2013}\right)^2=2013\\ \Leftrightarrow P\left(\sqrt{2013}-1\right)+6=2013\\ \Leftrightarrow P\left(\sqrt{2013}-1\right)=2007\)

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