Question

if
$\alpha \: and \beta \: are \: the \: zeros \: of \: the \: of \: the \: polynomial \: {x}^{2} - x - k \: and \: \alpha - \beta = - 7 \: find \: k$

Answer 1

$from \: \: the \: \: polynomial \\ \alpha + \beta = 1 \: \: and \: \: \alpha \beta = - k \\ given \: \: \alpha - \beta = - 7 \\ \\ using \: \: the \: \: identity \\ {( \alpha + \beta )}^{2} - {( \alpha - \beta )}^{2} = 4 \alpha \beta \\ {1}^{2} - {( - 7 )}^{2} = 4 \times - k \\ - 48 = - 4k \\ k = 12$