1.CM đẳng thức: \(\sqrt[3]{\sqrt[3]{2}-1}=\sqrt[3]{\frac{1}{9}}+\sqrt[3]{\frac{2}{9}}+\sqrt[3]{\frac{4}{9}}\)
2. Giai hệ phương trình:
\(\left\{{}\begin{matrix}x^2\left(2013y-2012\right)=1\\x\left(y^2+2012\right)=2013\end{matrix}\right.\)
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NV
Nguyễn Việt Lâm
Giáo viên
26 tháng 7 2020
ĐKXĐ: ...
Đặt \(\left\{{}\begin{matrix}\frac{1}{\sqrt[3]{x}}=a\\\frac{1}{\sqrt[3]{y}}=b\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}a^3+b^3=9\\\left(a+b\right)\left(a+1\right)\left(b+1\right)=18\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left(a+b\right)^3-3ab\left(a+b\right)=9\\\left(a+b\right)\left(ab+a+b+1\right)=18\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left(a+b\right)^3-3ab\left(a+b\right)=9\\ab\left(a+b\right)+\left(a+b\right)^2+\left(a+b\right)=18\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left(a+b\right)^3-3ab\left(a+b\right)=9\\3ab\left(a+b\right)+3\left(a+b\right)^2+3\left(a+b\right)=54\end{matrix}\right.\)
\(\Rightarrow\left(a+b\right)^3+3\left(a+b\right)^2+3\left(a+b\right)=63\)
\(\Leftrightarrow\left(a+b\right)^3+3\left(a+b\right)^2+3\left(a+b\right)+1=64\)
\(\Leftrightarrow\left(a+b+1\right)^3=4^3\)
\(\Leftrightarrow a+b+1=4\Rightarrow a+b=3\)
\(\Rightarrow3\left(ab+3+1\right)=18\Rightarrow ab=2\)
Theo Viet đảo; a và b là nghiệm:
\(t^2-3t+2=0\Rightarrow\left[{}\begin{matrix}t=1\\t=2\end{matrix}\right.\)
\(\Rightarrow\left(a;b\right)=\left(1;2\right);\left(2;1\right)\Rightarrow\left(x;y\right)=\left(1;\frac{1}{8}\right);\left(\frac{1}{8};1\right)\)
12 tháng 12 2023
\(\left\{{}\begin{matrix}\left|x-2\right|+2\sqrt{y+3}=9\\x+\sqrt{y+3}=-1\end{matrix}\right.\left(1\right)\)
ĐKXĐ: y>=-3
TH1: x>=2
Hệ phương trình(1) sẽ trở thành:
\(\left\{{}\begin{matrix}x-2+2\sqrt{y+3}=9\\x+\sqrt{y+3}=-1\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}x+2\sqrt{y+3}=11\\x+\sqrt{y+3}=-1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\sqrt{y+3}=12\\x+\sqrt{y+3}=-1\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}y+3=144\\x+12=-1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=144\\x=-13\left(loại\right)\end{matrix}\right.\)
=>Loại
TH2: x<2
hệ phương trình (1) sẽ trở thành \(\left\{{}\begin{matrix}-x+2+2\sqrt{y+3}=9\\x+\sqrt{y+3}=-1\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}-x+2\sqrt{y+3}=7\\x+\sqrt{y+3}=-1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}3\sqrt{y+3}=6\\x+\sqrt{y+3}=-1\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}\sqrt{y+3}=2\\x+2=-1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y+3=4\\x=-3\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}x=-3\\y=1\end{matrix}\right.\left(nhận\right)\)
13 tháng 2 2020
ĐKXĐ:...
a) \(\left\{{}\begin{matrix}\frac{x}{2}=\frac{y}{3}\\\frac{x+8}{y+4}=\frac{9}{4}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=\frac{2y}{3}\\\frac{\frac{2y}{3}+8}{y+4}=\frac{9}{4}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y=\frac{-12}{19}\\x=\frac{-8}{19}\end{matrix}\right.\)
Vậy...
b) \(\left\{{}\begin{matrix}0,75x-3,2y=10\\x\sqrt{3}-y\sqrt{2}=4\sqrt{3}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=\frac{3,2y+10}{0,75}\\\frac{\left(3,2y+10\right)\sqrt{3}}{0,75}-y\sqrt{2}=4\sqrt{3}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\frac{\frac{16\sqrt{3}}{5}y+10\sqrt{3}-\frac{3\sqrt{2}}{4}y}{0,75}=4\sqrt{3}\\x=\frac{3,2y+10}{0,75}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y\left(\frac{16\sqrt{3}}{5}-\frac{3\sqrt{2}}{4}\right)+10\sqrt{3}=3\sqrt{3}\\x=\frac{3,2y+10}{0,75}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y=\frac{-140\sqrt{3}}{64\sqrt{3}-15\sqrt{2}}\\x=\frac{\frac{-448\sqrt{3}}{64\sqrt{3}-15\sqrt{2}}+10}{0,75}\end{matrix}\right.\)
Nghiệm đẹp lắm.
c) \(\left\{{}\begin{matrix}\frac{2x+3}{y-1}=\frac{4x+1}{2y+1}\\\frac{x+2}{y-1}=\frac{x-4}{y+2}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left(2x+3\right)\left(2y+1\right)-\left(y-1\right)\left(4x+1\right)=0\\\left(x+2\right)\left(y+2\right)-\left(y-1\right)\left(x-4\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}6x+5y+4=0\\3x+6y=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=-2y\\-12y+5y+4=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y=\frac{4}{7}\\x=\frac{-8}{7}\end{matrix}\right.\)
Vậy...
10 tháng 1 2021
1) Ta có: \(\left\{{}\begin{matrix}3\sqrt{x}-\sqrt{y}=5\\2\sqrt{x}+3\sqrt{y}=18\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}9\sqrt{x}-3\sqrt{y}=15\\2\sqrt{x}+3\sqrt{y}=18\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}11\sqrt{x}=33\\3\sqrt{x}-\sqrt{y}=5\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{x}=3\\\sqrt{y}=4\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=9\\y=16\end{matrix}\right.\)
Vậy: Hệ phương trình có nghiệm duy nhất là \(\left\{{}\begin{matrix}x=9\\y=16\end{matrix}\right.\)
2) Ta có: \(\left\{{}\begin{matrix}\sqrt{x+3}-2\sqrt{y+1}=2\\2\sqrt{x+3}+\sqrt{y+1}=4\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}-2\sqrt{x+3}+4\sqrt{y+1}=-4\\2\sqrt{x+3}+\sqrt{y+1}=4\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}5\sqrt{y+1}=0\\\sqrt{x+3}-2\sqrt{y+1}=2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{y+1}=0\\\sqrt{x+3}=2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y+1=0\\x+3=4\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=-1\\x=1\end{matrix}\right.\)
Vậy: Hệ phương trình có nghiệm duy nhất là \(\left\{{}\begin{matrix}x=1\\y=-1\end{matrix}\right.\)
29 tháng 4 2023
4. Đk: \(x,y\ge0\)
\(\left\{{}\begin{matrix}\sqrt{x}+\sqrt{y+1}=1\\\sqrt{y}+\sqrt{x+1}=1\end{matrix}\right.\left(1\right)\)
Ta có: \(\left\{{}\begin{matrix}\sqrt{x}+\sqrt{y+1}\ge0+1=1\\\sqrt{y}+\sqrt{x+1}\ge0+1=1\end{matrix}\right.\left(2\right)\)
\(\left(1\right),\left(2\right)\Rightarrow\left\{{}\begin{matrix}\sqrt{x}=0,\sqrt{x+1}=1\\\sqrt{y}=0,\sqrt{y+1}=1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=0\\y=0\end{matrix}\right.\)<tmđk>
Vậy hệ pt có nghiệm \(\left(x,y\right)=\left(0;0\right)\)
DH
25 tháng 1 2020
\(2,\left\{{}\begin{matrix}x^3-2x^2y-15x=6y\left(2x-5-4y\right)\left(1\right)\\\frac{x^2}{8y}+\frac{2x}{3}=\sqrt{\frac{x^3}{3y}+\frac{x^2}{4}}-\frac{y}{2}\left(2\right)\end{matrix}\right.\)
\(\left(1\right)\Leftrightarrow\left(2y-x\right)\left(x^2-12y-15\right)=0\)\(\Leftrightarrow\left[{}\begin{matrix}2y=x\\y=\frac{x^2-15}{12}\end{matrix}\right.\)
Ta xét các trường hợp sau:
Trường hợp 1:
\(y=\frac{x^2-15}{12}\) thay vào phương trình \(\left(2\right)\) ta được:
\(\frac{3x^2}{2\left(x^2-15\right)}+\frac{2x}{3}=\sqrt{\frac{4x^3}{x^2-15}+\frac{x^2}{4}}-\frac{x^2-15}{24}\)
\(\Leftrightarrow\frac{36x^2}{x^2-15}-12\sqrt{\frac{x^2}{x^2-15}\left(x^2+16x-15\right)}+\left(x^2+16x-15\right)=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2+16x-15\ge0\\6\sqrt{\frac{x^2}{x^2-15}}=\sqrt{\left(x^2+16x-15\right)}\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x^2+16x-15\ge0\\36\frac{x^2}{x^2-15}=x^2+16x-15\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2+16x-15\ge0\\36x^2=\left(x^2-15\right)\left(x^2+16x-15\right)\left(3\right)\end{matrix}\right.\)
Ta xét phương trình \(\left(3\right):36x^2=\left(x^2-15\right)\left(x^2+16x-15\right)\)
Vì: \(x=0\) Không phải là nghiệm. Ta chia cả hai vế p.trình cho \(x^2\) ta được:
\(36=\left(x-\frac{15}{x}\right)\left(x+16-\frac{15}{x}\right)\)
Đặt: \(x-\frac{15}{x}=t\Rightarrow t^2+16t-36=0\Leftrightarrow\left[{}\begin{matrix}t=2\\t=-18\end{matrix}\right.\)
+ Nếu như:
\(t=2\Leftrightarrow x-\frac{15}{x}=2\Leftrightarrow x^2-2x-15=0\Leftrightarrow\left[{}\begin{matrix}x=5\\x=-3\end{matrix}\right.\)\(\Leftrightarrow x=5\)
+ Nếu như:
\(t=-18\Leftrightarrow x-\frac{15}{x}=-18\Leftrightarrow x^2+18x-15=0\Leftrightarrow\left[{}\begin{matrix}x=-9-4\sqrt{6}\\x=-9+4\sqrt{6}\end{matrix}\right.\Leftrightarrow x=-9-4\sqrt{6}\)
Trường hợp 2:
\(x=2y\) thay vào p.trình \(\left(2\right)\) ta được:
\(\Leftrightarrow\frac{x^2}{4x}+\frac{2x}{3}=\sqrt{\frac{2x^3}{3x}+\frac{x^2}{4}}-\frac{x}{4}\Leftrightarrow\frac{7}{6}x=\sqrt{\frac{11x^2}{12}}\Leftrightarrow x=0\left(ktmđk\right)\)
Vậy nghiệm của hệ đã cho là: \(\left(x,y\right)=\left(5;\frac{5}{6}\right),\left(-9-4\sqrt{6};\frac{27+12\sqrt{6}}{2}\right)\)
25 tháng 1 2020
Năm mới chắc bị lag @@ tớ sửa luôn đề câu 3 nhé :v
3, \(\left\{{}\begin{matrix}8\left(x^2+y^2\right)+4xy+\frac{5}{\left(x+y\right)^2}=13\left(1\right)\\2xy+\frac{1}{x+y}=1\left(2\right)\end{matrix}\right.\)
\(\left(1\right)\Leftrightarrow8\left[\left(x+y\right)^2-2xy\right]+4xy+\frac{5}{\left(x+y\right)^2}=13\)
Đặt \(\left\{{}\begin{matrix}x+y=a\\xy=b\end{matrix}\right.\)
\(\left(1\right)\Leftrightarrow8\left(a^2-2b\right)+4b+\frac{5}{a^2}=13\)
\(\Leftrightarrow8a^2-12b+\frac{5}{a^2}=13\)
Ta cũng có \(\left(2\right)\Leftrightarrow2b+\frac{1}{a}=1\)
\(\Leftrightarrow2b=1-\frac{1}{a}\)
Thay vào (1) ta được :
\(8a^2+\frac{5}{a^2}-6\cdot\left(1-\frac{1}{a}\right)=13\)
\(\Leftrightarrow8a^2+\frac{5}{a^2}-6+\frac{6}{a}=13\)
\(\Leftrightarrow8a^2+\frac{5}{a^2}+\frac{6}{a}=19\)
Giải pt được \(a=1\)
Khi đó \(b=\frac{1-\frac{1}{1}}{2}=0\)
Ta có hệ :
\(\left\{{}\begin{matrix}x+y=1\\xy=0\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x=0\\y=1\end{matrix}\right.\\\left\{{}\begin{matrix}x=1\\y=0\end{matrix}\right.\end{matrix}\right.\)
Vậy...