Question

If alpha and beta are the zeroes of the polynomial f(x) = 2x² - 7x + 3, find the value of Alpha^{2 +} Beta²

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{p}\left(\mathrm{x}\right)=3{\mathrm{x}}^2-14\mathrm{x}+15 \\ \mathrm{Now},\mathrm{\alpha }+\mathrm{\beta }=-\frac{\mathrm{coefficient}\mathrm{of}\mathrm{x}}{\mathrm{coefficient}\mathrm{of}{\mathrm{x}}^2}=-\frac{-14}{3}=\frac{14}{3} \\ \mathrm{\alpha \beta }=\frac{\mathrm{constant}\mathrm{term}}{\mathrm{coefficient}\mathrm{of}{\mathrm{x}}^2}=\frac{15}{3}=5 \\ \mathrm{Now},{\mathrm{\alpha }}^2+{\mathrm{\beta }}^2={\left(\mathrm{\alpha }+\mathrm{\beta }\right)}^2-2\mathrm{\alpha \beta } \\ ={\left(\frac{14}{3}\right)}^2-2\times 5 \\ =\frac{196}{9}-10 \\ =\frac{196-90}{9} \\ =\frac{106}{9} \end{gathered}$$