slt by Phúc dz
\(x^2-3y^2+2xy-2x+6y+8=0\)
\(\Leftrightarrow3y^2-x^2-2xy+2x-6y-8=0\)
\(\Leftrightarrow3y^2-y\left(2x+6\right)-\left(x^2-2x+8\right)\ge0\)
\(\Delta=\left(2x+6\right)^2-12\left(x^2-2x+8\right)\ge0\)
\(\Leftrightarrow-8x^2+48x-60\ge0\Leftrightarrow\frac{6-\sqrt{6}}{2}\le x\le\frac{6+\sqrt{6}}{2}\)
\(\Rightarrow1\le x\le4\)
Cho \(\left\{{}\begin{matrix}x,y,z>0\\x+2y+3z=3\end{matrix}\right.\)
Tìm MaxP biết:
\(P=\dfrac{88y^3-x^3}{2xy+16y^2}+\dfrac{297z^3-8y^3}{6yz+36z^2}+\dfrac{11x^3-27z^3}{3xz+4x^2}\)
Đặt 2y=a, 3z=b \(\Rightarrow x+a+b=3\)
\(\Rightarrow P=\dfrac{11a^3-x^3}{ax+4a^2}+\dfrac{11b^3-a^3}{ab+4b^2}+\dfrac{11x^3-b^3}{bx+4x^2}\)
Ta chứng minh bđt sau:
\(\dfrac{11a^3-x^3}{ax+4a^2}\le3a-x\Leftrightarrow11a^3-x^3\le\left(3a-x\right)\left(ax+4a^2\right)\Leftrightarrow11a^3-x^3\le12a^3+3a^2x-ax^2-4a^2x\Leftrightarrow a^3-a^2x-ax^2+x^3\ge0\Leftrightarrow a^2\left(a-x\right)-x^2\left(a-x\right)\ge0\Leftrightarrow\left(a-x\right)^2\left(a+x\right)\ge0\left(luondung\right)\)tương tự:
\(\dfrac{11x^3-b^3}{bx+4x^2}\le3x-b,\dfrac{11b^3-a^3}{ab+4b^2}\le3b-a\)
\(\Rightarrow P\le3\left(x+a+b\right)-\left(a+b+x\right)=2\left(a+b+x\right)=2.3=6\)
\(MaxP=6\Leftrightarrow x=1,y=\dfrac{1}{2},z=\dfrac{1}{3}\)
giải bất phương trình \(\left(\sqrt{13}-\sqrt{2x^2-2x+5}-\sqrt{2x^2-4x+4}\right)\left(x^6-x^3+x^2-x+1\right)\ge0\)
Giải PT và HPT:
1)\(\left\{{}\begin{matrix}xy+x+y=3\\\frac{1}{x^2+2x}+\frac{1}{y^2+2y}=\frac{2}{3}\end{matrix}\right.\)
2)\(\left(\sqrt{x+4}-2\right)\left(\sqrt{4-x}+2\right)=2x\)
3)\(\left\{{}\begin{matrix}xy\left(x+y\right)=2\\9xy\left(3x-y\right)+6=26x^3-2y^3\end{matrix}\right.\)
4)\(\left\{{}\begin{matrix}x^2-2xy+x-2y+3=0\\y^2-x^2+2xy+2x-2=0\end{matrix}\right.\)
Giải các pt sau:
a) \(\sqrt{x+8}+\frac{9x}{\sqrt{x+8}}-6\sqrt{x}=0\)
b) \(x^4-2x^3+\sqrt{2x^3+x^2+2}-2=0\)
c) \(3x\sqrt[3]{x+7}\left(x+\sqrt[3]{x+7}\right)=7x^3+12x^2+5x-6\)
d) \(4x^2+\left(8x-4\right)\sqrt{x}-1=3x+2\sqrt{2x^2+5x-3}\)
e) \(16x^2+19x+7+4\sqrt{-3x^2+5x+2}=\left(8x+2\right)\left(\sqrt{2-x}+2\sqrt{3x+1}\right)\)
f) \(\left(5x+8\right)\sqrt{2x-1}+7x\sqrt{x+3}=9x+8-\left(x+26\right)\sqrt{x-1}\)
g) \(\sqrt[3]{3x+1}+\sqrt[3]{5-x}+\sqrt[3]{2x-9}-\sqrt[3]{4x-3}=0\)
Giải hệ pt
1/\(\left\{{}\begin{matrix}4x\sqrt{y+1}+8x=\left(4x^2-4x-3\right)\sqrt{x+1}\\\dfrac{x}{x+1}+x^2=\left(y+2\right)\sqrt{\left(x+1\right)\left(y+1\right)}\end{matrix}\right.\)
2/\(\left\{{}\begin{matrix}x\sqrt{y^2+6}+y\sqrt{x^2+3}=7xy\\x\sqrt{x^2+3}+y\sqrt{y^2+6}=x^2+y^2+2\end{matrix}\right.\)\(\left\{{}\begin{matrix}x\sqrt{y^2+6}+y\sqrt{x^2+3}=7xy\\x\sqrt{x^2+3}+y\sqrt{y^2+6}=x^2+y^2+2\end{matrix}\right.\)
3/\(\left\{{}\begin{matrix}\left(2x+y-1\right)\left(\sqrt{x+3}+\sqrt{xy}+\sqrt{x}\right)=8\sqrt{x}\\\left(\sqrt{x+3}+\sqrt{xy}\right)^2+xy=2x\left(6-x\right)\end{matrix}\right.\)\(\left\{{}\begin{matrix}\left(2x+y-1\right)\left(\sqrt{x+3}+\sqrt{xy}+\sqrt{x}\right)=8\sqrt{x}\\\left(\sqrt{x+3}+\sqrt{xy}\right)^2+xy=2x\left(6-x\right)\end{matrix}\right.\)
4/\(\left\{{}\begin{matrix}\sqrt{xy+x+2}+\sqrt{x^2+x}-4\sqrt{x}=0\\xy+x^2+2=x\left(\sqrt{xy+2}+3\right)\end{matrix}\right.\)\(\left\{{}\begin{matrix}\sqrt{xy+x+2}+\sqrt{x^2+x}-4\sqrt{x}=0\\xy+x^2+2=x\left(\sqrt{xy+2}+3\right)\end{matrix}\right.\)
m.n giúp e mấy bài này vs ạ!!
giải hệ phương trình:
1, \(\left\{{}\begin{matrix}2y\left(4y^2+3x^2\right)=x^4\left(x^2+3\right)\\2012^x\left(\sqrt{2y-2x+5}-x+1\right)=4024\end{matrix}\right.\)
2, \(\left\{{}\begin{matrix}x^3-2x^2y-15x=6y\left(2x-5-4y\right)\\\frac{x^2}{8y}+\frac{2x}{3}=\sqrt{\frac{x^3}{3y}+\frac{x^2}{4}}-\frac{y}{2}\end{matrix}\right.\)
3, \(\left\{{}\begin{matrix}8\left(x^2+y^2\right)+4xy+\frac{5}{\left(x+y\right)^2}=13\\2x+\frac{1}{x+y}=1\end{matrix}\right.\)
Giải hệ:
\(\left\{{}\begin{matrix}x^2+y^2+xy=5\\27x^3+6y^2x=2+y^3+30x^2y\end{matrix}\right.\)
\(\left\{{}\begin{matrix}x^2+y^2+\frac{8xy}{x+y}=16\\\frac{x^2}{8y}+\frac{2x}{3}=\sqrt{\frac{x^3}{3y}+\frac{x^2}{4}}-\frac{y}{2}\end{matrix}\right.\), \(\left\{{}\begin{matrix}\frac{1}{3x}+\frac{2x}{3y}=\frac{x+\sqrt{y}}{2x^2+y}\\2\left(2x+\sqrt{y}\right)=\sqrt{2x+6}-y\end{matrix}\right.\)
\(\left\{{}\begin{matrix}x^2y-3x-1=3x\sqrt{y}\left(\sqrt{1-x}-1\right)^3\\\sqrt{8x^2-3xy+4y^2}+\sqrt{xy}=4y\end{matrix}\right.\)
Cho các số a,b,c là các số k âm sao cho tổng hai số bất kì đều dương.CMR \(\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}+\frac{16\sqrt{ab+bc+ac}}{a+b+c}\ge8\)
Giải phương trình vô tỉ:
a) \(4x^2-4x-10=\sqrt{8x^2-6x-10}\)
b) \(\sqrt{\left(x+1\right)\left(2-x\right)}=1+2x-2x^2\)
c) \(\sqrt{3x+8+6\sqrt{3x-1}}+\sqrt{3x+8-6\sqrt{3x-1}}=3x+4\)
d) \(2x\sqrt{x^2-x+1}+4\sqrt{3x+1}=2x^2+2x+6\)
1, Giải các hệ phương trình sau
a, \(\left\{{}\begin{matrix}\left(x+y\right)^2-2xy=26\\x+y=6\end{matrix}\right.\)
b,\(\left\{{}\begin{matrix}2x^2+x-y=0\\xy+3y-5x=7\end{matrix}\right.\)
c, \(\left\{{}\begin{matrix}\left(x-1\right)^2=1-y\\\left(x^2-y\right)^2=2xy\left(1+x\right)\end{matrix}\right.\)
d, \(\left\{{}\begin{matrix}x^2y+y^2x=2\\x^3+y^3+6=8x^2y^2\end{matrix}\right.\)
