Find the roots of the given equation:

contexto.140

3 years ago

$\begin{array}{l} \frac{1}{2 a+b+2 x}=\frac{1}{2 a}+\frac{1}{b}+\frac{1}{2 x} \\ \Rightarrow \frac{1}{2 a+b+2 x}-\frac{1}{2 x}=\frac{1}{2 a}+\frac{1}{b} \\ \Rightarrow \frac{2 x-2 a-b-2 x}{2 x(2 a+b+2 x)}=\frac{2 a+b}{2 a b} \\ \Rightarrow \frac{-(2 a+b)}{4 x^{2}+4 a x+2 b x}=\frac{2 a+b}{2 a b} \\ \Rightarrow 4 x^{2}+4 a x+2 b x=-2 a b \\ \Rightarrow 4 x^{2}+4 a x+2 b x+2 a b=0 \\ \Rightarrow 4 x(x+a)+2 b(x+a)=0 \\ \Rightarrow(x+a)(4 x+2 b)=0 \\ \Rightarrow x+a=0 \text { or } 4 x+2 b=0 \\ \Rightarrow x=-a \text { or } x=-\frac{b}{2} \end{array}$

Hence, and are the roots of the give equation.