Question

If α and β are the zeroes of a polynomial f(x) = x² + x − 2, find the value of $\left(\frac{1}{\alpha }-\frac{1}{\beta }\right)$

Answer 1

By using the relationship between the zeroes of the quadratic ploynomial.
We have,
Sum of zeroes = $\frac{-\left(\mathrm{coefficient}\mathrm{of}\mathit{}x\right)}{\mathrm{coefficent}\mathrm{of}{x}^2}$ and Product of zeroes = $\frac{\mathrm{constant}\mathrm{term}}{\mathrm{coefficent}\mathrm{of}{x}^2}$

$$\begin{gathered} \Rightarrow \alpha +\beta =\frac{-1}{1}\mathrm{and}\alpha \beta =\frac{-2}{1} \\ \Rightarrow \alpha +\beta =-1\mathrm{and}\alpha \beta =-2 \\ \mathrm{Now},{\left(\frac{1}{\alpha }-\frac{1}{\beta }\right)}^2={\left(\frac{\beta -\alpha }{\alpha \beta }\right)}^2 \end{gathered}$$

$$\begin{gathered} =\frac{{\left(\alpha +\beta \right)}^2-4\alpha \beta }{{\left(\alpha \beta \right)}^2}\left[\because {\left(\beta -\alpha \right)}^2={\left(\alpha +\beta \right)}^2-4\alpha \beta \right] \\ =\frac{{\left(-1\right)}^2-4\left(-2\right)}{{\left(-2\right)}^2}\left[\because \alpha +\beta =-1\mathrm{and}\alpha \beta =-2\right] \\ =\frac{{\left(-1\right)}^2-4\left(-2\right)}{4} \\ =\frac{9}{4} \end{gathered}$$

$$\begin{gathered} \because {\left(\frac{1}{\alpha }-\frac{1}{\beta }\right)}^2=\frac{9}{4} \\ \Rightarrow \frac{1}{\alpha }-\frac{1}{\beta }=\pm \frac{3}{2} \end{gathered}$$