Question

If α and β are zeroes of polynomial p(x)= 4x^{2} -5x - 1 then find: 1/α^{2} - 1/β^{2}

Answer 1

${ \bold{ \green{ \underline{ANSWER} : - } }}$

${ \bold{ \underline{Given \: \: polynomial} : - }} \\ \\ { \bold{ \pink{p(x) = 4 {x}^{2} - 5x - 1}}}$

${ \bold{ \underline{ \underline{To \: find}} : - }} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: { \bold{ \orange{ \frac{1}{ { \alpha }^{2} } - \frac{1}{ { \beta }^{2} } }}}$

${ \bold{ \pink{ = \frac{1}{ { \alpha }^{2} } - \frac{1}{ { \beta }^{2} } }}} \\ \\ { \bold{ \pink{ = \frac{ { \beta }^{2} - { \alpha }^{2} }{ {( \alpha \beta )}^{2} } }}} \\ \\ { \bold{ \pink{ = \frac{ - ( \alpha + \beta )( \alpha - \beta )}{ {( \alpha \beta )}^{2} } }}} \\ \\ { \bold{ \pink{ = \frac{ - ( - \frac{b}{a})( \frac{ \sqrt{d} }{a}) }{ {( \frac{c}{a} )}^{2} } }}} \\ \\ { \bold{ \pink{ = \frac{b \sqrt{d} }{ {c}^{2} } }}} \\ \\ { \bold{ \pink{ = \frac{ (- 5) \sqrt{ {( - 5)}^{2} - 4(4)( - 1)} }{1} }}} \\ \\ { \bold{ \pink{ = - 5 \sqrt{25 + 16} }}} \\ \\ { \bold{ \pink{ = - 5 \sqrt{41} }}} \\ \\ { \bold{ \boxed{\pink{ \huge{FOLLOW \: \: ME... }}}}}$