\(\frac{1}{x}+\frac{1}{x+3}=\frac{1}{2}\)
\(\Leftrightarrow\frac{2\left(x+3\right)}{2x\left(x+3\right)}+\frac{2x}{2x\left(x+3\right)}=\frac{x\left(x+3\right)}{2x\left(x+3\right)}\)
\(\Leftrightarrow2x+6+2x=x^2+3x\)
\(\Leftrightarrow x=3\)
\(\frac{1}{x}+\frac{1}{x+3}=\frac{1}{2}\)
\(\frac{1}{x+x+3}=\frac{1}{2}\)
x+x+3=2
2x=-1
x=-1/2
Theo đầu bài ta có:
\(\frac{1}{x}+\frac{1}{x+3}=\frac{1}{2}\)
\(\Rightarrow\frac{\left(x+3\right)+x}{x\left(x+3\right)}=\frac{1}{2}\)
\(\Rightarrow\frac{2x+3}{x^2+3x}=\frac{1}{2}\)
\(\Rightarrow2\cdot\left(2x+3\right)=x^2+3x\)
\(\Rightarrow4x+6=x^2+3x\)
\(\Rightarrow6=x^2+3x-4x\)
\(\Rightarrow x\left(x-1\right)=6\)
Ta thấy: \(\hept{\begin{cases}x-\left(x-1\right)=1\\x+\left(x-1\right)=a\end{cases}}\Rightarrow\hept{\begin{cases}x=\frac{a+1}{2}\\x-1=\frac{a-1}{2}\end{cases}}\) ( với a là hằng số )
\(\Rightarrow x\left(x+1\right)=\frac{a+1}{2}\cdot\frac{a-1}{2}\)
\(\Rightarrow\frac{\left(a+1\right)\left(a-1\right)}{4}=6\)
\(\Rightarrow a^2-1=24\)
\(\Rightarrow a^2=25\)
\(\Rightarrow\hept{\begin{cases}a=5\\a=-5\end{cases}}\)
- Nếu a = 5 thì x = ( 5 + 1 ) : 2 = 3
- Nếu a = -5 thì x = ( -5 + 1 ) : 2 = -2
