Question

α, β are roots of y² – 2y –7 = 0 find,
(1) α² + β²
(2) α³ + β³

Answer 1

α, β are roots of y² – 2y –7 = 0
$a=1,b=-2,c=-7$$a=1,b=-2,c=-7$
(1)
$$\begin{gathered} \alpha +\beta =\frac{-b}{a}=\frac{-\left(-2\right)}{1}=2 \\ \alpha \beta =\frac{c}{a}=\frac{-7}{1}=-7 \end{gathered}$$
$$\begin{gathered} {\alpha }^2+{\beta }^2={\left(\mathrm{\alpha }+\mathrm{\beta }\right)}^2-2\alpha \beta \\ ={\left(2\right)}^2-2\left(-7\right) \\ =4+14 \\ =18 \end{gathered}$$

(2)
$$\begin{gathered} {\alpha }^3+{\beta }^3={\left(\alpha +\beta \right)}^3-3\alpha \beta \left(\alpha +\beta \right) \\ ={\left(2\right)}^3-3\left(-7\right)\left(2\right) \\ =8+42 \\ =50 \end{gathered}$$