Question

Solve each of the following quadratic equations:

$\frac{1}{x+1}+\frac{2}{x+2}=\frac{5}{x+4},x\ne -1,-2,-4$ [CBSE 2013C]

Answer 1

$$\begin{gathered} \frac{1}{x+1}+\frac{2}{x+2}=\frac{5}{x+4},x\ne -1,-2,-4 \\ \Rightarrow \frac{x+2+2x+2}{\left(x+1\right)\left(x+2\right)}=\frac{5}{x+4} \\ \Rightarrow \frac{3x+4}{x^2+3x+2}=\frac{5}{x+4} \\ \Rightarrow \left(3x+4\right)\left(x+4\right)=5\left(x^2+3x+2\right) \end{gathered}$$
$$\begin{gathered} \Rightarrow 3x^2+16x+16=5x^2+15x+10 \\ \Rightarrow 2x^2-x-6=0 \\ \Rightarrow 2x^2-4x+3x-6=0 \\ \Rightarrow 2x\left(x-2\right)+3\left(x-2\right)=0 \end{gathered}$$
$$\begin{gathered} \Rightarrow \left(x-2\right)\left(2x+3\right)=0 \\ \Rightarrow x-2=0\mathrm{or}2x+3=0 \\ \Rightarrow x=2\mathrm{or}x=-\frac{3}{2} \end{gathered}$$
Hence, 2 and $-\frac{3}{2}$ are the roots of the given equation.