Question

Solve the following quadratic equation by factorisation:
1/x - 1/x-2 = 3

Answer 1

$$\begin{gathered} \frac{1}{\mathrm{x}}-\frac{1}{\mathrm{x}-2}=3 \\ \Rightarrow \frac{\mathrm{x}-2-\mathrm{x}}{\mathrm{x}\left(\mathrm{x}-2\right)}=3 \\ \Rightarrow -\frac{2}{{\mathrm{x}}^2-2\mathrm{x}}=3 \\ \Rightarrow 3{\mathrm{x}}^2-6\mathrm{x}+2=0 \\ \mathrm{Now},\mathrm{a}=3;\mathrm{b}=-6;\mathrm{c}=2 \\ \mathrm{Now},\mathrm{D}={\mathrm{b}}^2-4\mathrm{ac}={\left(-6\right)}^2-4\left(3\right)\left(2\right)=36-24=12>0 \\ \mathrm{So},\mathrm{by}\mathrm{quadratic}\mathrm{formula}, \\ \mathrm{x}=\frac{-\mathrm{b}+\sqrt{\mathrm{D}}}{2\mathrm{a}}\mathrm{and}\mathrm{x}=\frac{-\mathrm{b}-\sqrt{\mathrm{D}}}{2\mathrm{a}} \\ \Rightarrow \mathrm{x}=\frac{6+\sqrt{12}}{6}\mathrm{and}\mathrm{x}=\frac{6-\sqrt{12}}{6} \\ \Rightarrow \mathrm{x}=\frac{6+2\sqrt{3}}{6}\mathrm{and}\mathrm{x}=\frac{6-2\sqrt{3}}{6} \\ \Rightarrow \mathrm{x}=\frac{3+\sqrt{3}}{3}\mathrm{and}\mathrm{x}=\frac{3-\sqrt{3}}{3} \end{gathered}$$