Question

please solve this question

Answer 1

${x}^{2} + x - 6 \\ let \: the \: zeros \: are \: \alpha \: and \: \beta \\ \alpha + \beta = - 1 \\ \alpha \beta = - 6$
$\frac{1}{ \alpha } + \frac{1}{ \beta } = \frac{ \alpha + \beta }{ \alpha \beta } = \frac{ - 1}{ - 6} = \frac{1}{6}$
${ \alpha }^{2} + { \beta }^{2} = { \alpha }^{2} + { \beta }^{2} + 2 \alpha \beta - 2 \alpha \beta \\ = ( { \alpha + \beta )}^{2} - 2 \alpha \beta \\ = ( { - 1)}^{2} - 2( - 6) \\ = 1 + 12 = 13$
$( { \alpha - \beta )}^{2} = ( { \alpha + \beta )}^{2} - 4 \alpha \beta \\ {( \alpha - \beta )}^{2} = {( - 1)}^{2} - 4( - 6) \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 1 + 24 = 25 \\ \\ ( \alpha - \beta ) = \sqrt{25} \\ ( \alpha - \beta ) = 5$