giải phương trình
\(\frac{3x+1}{2}-x=1\)
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\(\frac{3x}{x^2+x+1}+\frac{3x}{x^2-x+1}=4\)
Xét x=0 không phải là nghiệm của phương trình, chia cả tử và mẫu của mỗi phân thức ở VT cho x, ta được:
\(\frac{3}{x+1+\frac{1}{x}}+\frac{3}{x-1+\frac{1}{x}}=4\)
Đặt \(x+\frac{1}{x}=y,\)khi đó phương trình có dạng:
\(\frac{3}{y+1}+\frac{3}{y-1}=4\Leftrightarrow\frac{6y}{y^2-1}=4\)
\(\Rightarrow4y^2-4=6y\Leftrightarrow4y^2-6y-4=0\)
\(\Leftrightarrow4y^2-8y+2y-4=0\)
\(\Leftrightarrow4y\left(y-2\right)+2\left(y-2\right)=0\Leftrightarrow\left(y-2\right)\left(4y+2\right)=0\)
\(\Leftrightarrow\left(y-2\right)\left(2y+1\right)=0\Leftrightarrow\orbr{\begin{cases}y=2\\y=-\frac{1}{2}\end{cases}}\)
\(\Rightarrow\orbr{\begin{cases}x+\frac{1}{x}=2\Leftrightarrow x^2+1=2x\Leftrightarrow x^2-2x+1=0\\x+\frac{1}{x}=-\frac{1}{2}\Leftrightarrow x^2+1=-\frac{1}{2}x\Leftrightarrow x^2+\frac{1}{2}x+1=0\end{cases}}\)
\(\Leftrightarrow\orbr{\begin{cases}\left(x-1\right)^2=0\\x^2+2.x.\frac{1}{4}+\frac{1}{16}+\frac{15}{16}=0\end{cases}\Leftrightarrow}\orbr{\begin{cases}x=1\\\left(x+\frac{1}{4}\right)^2=-\frac{15}{16}\left(\times\right)\end{cases}}\)
Vậy pt có nghiệm duy nhất là x=1.
\(\frac{2}{x+\frac{1}{1+\frac{x+1}{x+2}}}=\frac{6}{3x-1}\)
\(\frac{2}{x+\frac{1}{\frac{x+2+x+1}{x+2}}}=\frac{6}{3x-1}\)
\(\frac{2}{x+\frac{1}{\frac{2x+3}{x+2}}}=\frac{6}{3x-1}\)
\(\frac{2}{x+\frac{x+2}{2x+3}}=\frac{6}{3x-1}\)
\(\frac{2}{\frac{2x+3+x+2}{2x+3}}=\frac{6}{3x-1}\)
\(\frac{2}{\frac{3x+5}{2x+3}}=\frac{6}{3x-1}\)
\(\frac{4x+6}{3x+5}=\frac{6}{3x-1}\)
\(\Rightarrow\left(4x+6\right)\left(3x-1\right)=6\left(3x+5\right)\)
\(\Rightarrow12x^2-4x+18x-6=18x+30\)
\(\Rightarrow12x^2-4x+18x-18x=30+6\)
\(\Rightarrow12x^2-4x-36=0\)
\(\Rightarrow3x^2-x-9=0\)
\(\Rightarrow x^2-\frac{1}{3}x-3=0\)
\(\Rightarrow x^2-2.\frac{1}{6}x+\frac{1}{36}-\frac{1}{36}-3=0\)
\(\Rightarrow\left(x-\frac{1}{6}\right)^2-\frac{109}{36}=0\)
\(\Rightarrow\left(x-\frac{1}{6}-\frac{\sqrt{109}}{6}\right)\left(x-\frac{1}{6}+\frac{\sqrt{109}}{6}\right)=0\)
\(\Rightarrow\orbr{\begin{cases}x=\frac{1+\sqrt{109}}{6}\\x=\frac{1-\sqrt{109}}{6}\end{cases}}\)
làm lại nhé, chỗ kia quy đồng sai
lần này làm theo cách khác
\(\frac{2}{x+\frac{1}{1+\frac{x+1}{x+2}}}=\frac{6}{3x-1}\)
\(\frac{2}{x+\frac{1}{\frac{x+2+x+1}{x+2}}}=\frac{2}{x-\frac{1}{3}}\)
\(\Rightarrow x+\frac{1}{\frac{2x+3}{x+2}}=x-\frac{1}{3}\)
\(\Rightarrow\frac{x+2}{2x+3}=\frac{-1}{3}\)
\(\Rightarrow\left(x+2\right).3=-1.\left(2x+3\right)\)
\(\Rightarrow3x+6=-2x-3\)
\(\Rightarrow3x+2x=-3-6\)
\(\Rightarrow5x=-9\)
\(\Rightarrow x=\frac{-9}{5}\)
vậy \(x=\frac{-9}{5}\)
Vừa lm xong mt bị sụp ...
\(\frac{1}{x-1}+\frac{3}{3x+5}=\frac{2}{x+2}+\frac{1}{x+3}\)ĐKXĐ : \(x\ne1;-\frac{5}{3};-2;-3\)
\(\frac{1}{x-1}+\frac{3}{3x+5}-\frac{2}{x+2}-\frac{1}{x+3}=0\)
\(\frac{\left(3x+5\right)\left(x+2\right)\left(x+3\right)}{\left(x-1\right)\left(3x+5\right)\left(x+2\right)\left(x+3\right)}+\frac{3\left(x-1\right)\left(x+2\right)\left(x+3\right)}{\left(3x+5\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}-\frac{2\left(x-1\right)\left(3x+5\right)\left(x+3\right)}{\left(x+2\right)\left(x-1\right)\left(3x-5\right)\left(x+3\right)}-\frac{\left(x-1\right)\left(3x+5\right)\left(x+2\right)}{\left(x+3\right)\left(x-1\right)\left(3x+5\right)\left(x+2\right)}=0\)
Khử mẫu và rút gọn ta đc : \(-3x^3+2x^2+45x+52=0\)
Mời cao nhân giải tiếp.
\(\frac{1}{x-1}+\frac{6}{3x+5}=\frac{2}{x+2}+\frac{1}{x+3}\)
\(\Leftrightarrow\frac{3x+5+6x-6}{3x^2+2x-5}=\frac{2x+6+x+2}{x^2+5x+6}\)
\(\Leftrightarrow\frac{9x-1}{3x^2+2x-5}=\frac{3x+8}{x^2+5x+6}\)
\(\Rightarrow9x^3+44x^2+49x-6=9x^3+30x^2+x-40\)
\(\Leftrightarrow14x^2-48x+34=0\)
\(\Rightarrow14x^2-14x-34x+34=0\)
\(\Rightarrow\left(x-1\right)\left(14x-34\right)=0\)
\(\Rightarrow\orbr{\begin{cases}x-1=0\\14x-34=0\end{cases}\Rightarrow\orbr{\begin{cases}x=1\\x=\frac{17}{7}\end{cases}}}\)
Ngu nên làm dài dòng thôi
\(\frac{1}{2-x}+1=\frac{1}{x+2}-\frac{6-x}{3x^2-12}\)ĐKXĐ : \(x\ne\pm2\)
\(\Leftrightarrow\frac{-3\left(x+2\right)}{3\left(x-2\right)\left(x+2\right)}+\frac{3\left(x-2\right)\left(x+2\right)}{3\left(x-2\right)\left(x+2\right)}=\frac{3\left(x-2\right)}{3\left(x-2\right)\left(x+2\right)}+\frac{x-6}{3\left(x-2\right)\left(x+2\right)}\)
\(\Leftrightarrow\frac{-3x-6+3\left(x^2-4\right)}{3\left(x-2\right)\left(x+2\right)}-\frac{3x-6+x-6}{3\left(x-2\right)\left(x+2\right)}=0\)
\(\Leftrightarrow\frac{-3x-6+3x^2-12-3x+6-x+6}{3\left(x-2\right)\left(x+2\right)}=0\)
\(\Leftrightarrow\frac{-7x-6+3x^2}{3\left(x-2\right)\left(x+2\right)}=0\)
\(\Leftrightarrow3x^2-7x-6=0\)
\(\Leftrightarrow3x^2-9x+2x-6=0\)
\(\Leftrightarrow3x\left(x-3\right)+2\left(x-3\right)=0\)
\(\Leftrightarrow\left(x-3\right)\left(3x+2\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}x=3\\x=\frac{-2}{3}\end{cases}}\)( thỏa mãn )
Vậy....
\(\frac{3x+1}{2}-x=1\)
\(\Leftrightarrow\frac{3x+1}{2}-\frac{2x}{2}=\frac{2}{2}\)
\(\Rightarrow3x+1-2x=2\)
\(\Leftrightarrow3x-2x=2-1\)
\(\Leftrightarrow x=1\)
Vậy tập nghiệm của pt là S={1}