Question

Solve the following equations (x $\ge$ 0)

$\frac{x^2-5x+12}{5x-12}=\frac{x^2-x-4}{x+4}$

Answer 1

$$\begin{gathered} \frac{x^2-5x+12}{5x-12}=\frac{x^2-x-4}{x+4} \\ \frac{x^2-5x+12+5x-12}{5x-12}=\frac{x^2-x-4+x+4}{x+4}\left(\mathrm{Componendo}\right) \\ \Rightarrow \frac{x^2}{5x-12}=\frac{x^2}{x+4} \\ \mathrm{Now},\mathrm{for}x=0,\mathrm{the}\mathrm{equation}\mathrm{is}\mathrm{satisfied}. \\ \mathrm{Thus},x=0\mathrm{is}\mathrm{one}\mathrm{of}\mathrm{the}\mathrm{solutions}. \\ \mathrm{When}x\ne 0,x^2\ne 0. \\ \therefore \frac{1}{5x-12}=\frac{1}{x+4}(\mathrm{Dividing}\mathrm{by}{x}^2) \\ \Rightarrow x+4=5x-12 \\ \Rightarrow -4x=-16 \\ \Rightarrow x=4 \\ \mathrm{Hence},\mathrm{either}x=0\mathrm{or}x=4\mathrm{is}\mathrm{the}\mathrm{solution}. \end{gathered}$$