Question

If $\alpha$ & $\beta$ roots of the equation $a{x}^2+bx+c=0$ then find $(1+\alpha +{\alpha }^2)(1+\beta +{\beta }^2)$

Answer 1

$\alpha +\beta =-b/a$

$\alpha \beta =c/a$

$(1+\alpha +{\alpha }^2)(1+\beta +{\beta }^2)$

$=1+\beta +{\beta }^2+\alpha +\alpha \beta +\alpha {\beta }^2+{\alpha }^2+{\alpha }^2\beta +{\alpha }^2{\beta }^2$

$=1+(\alpha +\beta )+({\alpha }^2+{\beta }^2)+\alpha \beta +\alpha \beta (\alpha +\beta )+(\alpha \beta {)}^2$

$=1-b/2+(\alpha +\beta {)}^2-2\alpha \beta )+\alpha \beta +c/a(-b/a)+(c/a{)}^2$

$=1-\frac{b}{a}+\frac{b^2}{a^2}-\alpha \beta -\frac{bc}{a^2}+\frac{c^2}{a^2}$

$=1-\frac{b}{a}+\frac{b^2}{a^2}-\frac{c}{a}-\frac{bc}{a^2}+\frac{c^2}{a^2}$

$=1-\left(\frac{b+c}{a}\right)+\left(\frac{b^2-bc+c^2}{a^2}\right)$

$=\frac{a^2-ab-ac+b^2-bc+c^2}{a^2}=\frac{a^2+b^2+c^2-ab-bc-ca}{a^2}$.