If the equation 2x² - 7x + 12 = 0 has two roots $$\alpha$$ and $$\beta$$, then the value of $$\frac{\alpha }{\beta }{\text{ + }}\frac{\beta }{\alpha }\,{\text{is?}}$$

Solution(By Examveda Team)

$$\eqalign{ & {\text{2}}{x^2} - 7x + 12 = 0 \cr & {\text{roots are }}\alpha {\text{ , }}\beta \cr & \therefore \alpha \beta = + \frac{c}{a},\alpha + \beta = \frac{{ - b}}{a} \cr & \therefore \alpha + \beta = + \frac{7}{2},\alpha \beta = \frac{{12}}{2} = 6 \cr & \therefore \frac{\alpha }{\beta } + \frac{\beta }{\alpha } \cr & = \frac{{{\alpha ^2} + {\beta ^2}}}{{\alpha \beta }} \cr & = \frac{{{{\left( {\alpha + \beta } \right)}^2} - 2\alpha \beta }}{{\alpha \beta }} \cr & = \frac{{{{\left( {\frac{7}{2}} \right)}^2} - 2 \times 6}}{6} \cr & = \frac{{\frac{{49}}{4} - 12}}{6} \cr & = \frac{{49 - 48}}{{6 \times 4}} \cr & = \frac{1}{{24}} \cr} $$