If $\mathrm{\alpha },\mathrm{\beta }$ are the roots of x² – px + q = 0 then $\left(\mathrm{\alpha }+\mathrm{\beta }\right)\mathrm{x}-({\mathrm{\alpha }}^2+{\mathrm{\beta }}^2)\frac{{\mathrm{x}}^2}{2}+({\mathrm{\alpha }}^3+{\mathrm{\beta }}^3)\frac{{\mathrm{x}}^3}{3}-\dots \dots \infty =$

$\mathrm{\alpha }+\mathrm{\beta }=\mathrm{p}, \mathrm{\alpha \beta }=\mathrm{q}$

$$\begin{gathered} \left(\mathrm{\alpha }+\mathrm{\beta }\right)\mathrm{x}-({\mathrm{\alpha }}^2+{\mathrm{\beta }}^2)\frac{{\mathrm{x}}^2}{2}+({\mathrm{\alpha }}^3+{\mathrm{\beta }}^3)\frac{{\mathrm{x}}^3}{3}-\dots \dots \infty = \\ =\left(\mathrm{\alpha } \mathrm{x}-\frac{{(\mathrm{\alpha } \mathrm{x})}^2}{2}+\frac{{(\mathrm{\alpha } \mathrm{x})}^{36}}{3}-....\right) + \left(\mathrm{\beta x}-\frac{{(\mathrm{\beta } \mathrm{x})}^2}{2}+\frac{{(\mathrm{\beta } \mathrm{x})}^{36}}{3}-....\right) \\ =\log \left(1+\mathrm{\alpha } \mathrm{x}\right)+\log \left(1+\beta \mathrm{x}\right) \\ =\log \left(1+\left(\alpha +\beta \right)x+\alpha \beta x^2\right) \\ =\log \left(1+\left(p\right)x+q{x}^2\right) \end{gathered}$$