Question

If $\alpha ,\beta$ are the roots of the equation $a{x}^2+bx+c=0$,

Then t $\times$ ${\log }_e(a-bx+c{x}^2)={\log }_{e}a+(\alpha +\beta )x-\frac{1}{2}({\alpha }^2+{\beta }^2)x^2+\frac{1}{3}({\alpha }^3+{\beta }^3)x^3-...to\infty$ Find t

Answer 1

Since $\alpha$ and $\beta$ are the roots of the equation $a{x}^2+bx+c=0$
$\therefore$ $\alpha +\beta =-\frac{b}{a}$ and $\alpha \beta =\frac{c}{a}$
$\therefore$ $\log a+(\alpha +\beta )x-\left(\frac{{\alpha }^2+{\beta }^2}{2}\right)x^2+\left(\frac{{\alpha }^3+{\beta }^3}{3}\right)x^3-...$
$=\log a+(\alpha x-\frac{(\alpha x{)}^2}{2}+\frac{(\alpha x{)}^3}{3}-...)+(\beta x-\frac{(\beta x{)}^2}{2}+\frac{(\beta x{)}^3}{3}-...)$
$=\log a+\log (1+\alpha x)+\log (1+\beta x)$
$=\log a+\log (1+(\alpha +\beta )x+\alpha \beta x^2)$
$=\log (a-bx+c{x}^2)$