Question

If $\alpha$ and $\beta$ are the zeros of the polynomial $2x^2-4x+5$, find the value of
(i) ${\alpha }^2+{\beta }^2$
(ii) $\frac{1}{\alpha }+\frac{1}{\beta }$
(iii) $(\alpha -\beta {)}^2$
(iv) ${\alpha }^3+{\beta }^3$

Answer 1

$2x^2-4x+5$

$\alpha +\beta =\frac{-b}{a}=\frac{-(-4)}{2}=\frac{4}{2}=2$

$\alpha \beta =\frac{c}{a}=\frac{5}{2}$

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

$=4-5=-1$

$\frac{1}{\alpha }+\frac{1}{\beta }=\frac{\alpha +\beta }{\alpha \beta }=\frac{2}{\frac{5}{2}}=\frac{4}{5}$

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

$=-1-2\left(\frac{5}{2}\right)=-1-5$

$=-6$

$({\alpha }^3+{\beta }^3)=(\alpha +\beta )({\alpha }^2+{\beta }^2-\alpha \beta )$

$=\left(2\right)(-1\frac{-5}{2})$

$=2\left(\frac{-7}{2}\right)=-7$