Question

If alpha and beta are the zeroes of f(x)ax2 +bx+c find alpha2-beta2

Answer 1

$f(x) = a {x}^{2} + bx + c = 0 \\ \\ wkt \\ \\ \alpha + \beta = - \frac{b}{a} \\ \\ \alpha \beta = \frac{c}{a} \\ \\ \alpha - \beta = \frac{ \sqrt{ {b}^{2} - 4ac} }{a} \\ \\ {x}^{2} - {y}^{2} = (x + y)(x - y) \\ \\ now \\ \\ \bold{ \large{ { \alpha }^{2} - { \beta }^{2} }} \\ \\ \bold{ \large{ = ( \alpha + \beta )( \alpha - \beta )}} \\ \\ \bold{ \large{ = ( - \frac{b}{a} )( \frac{ \sqrt{ {b}^{2} - 4ac} }{a} )}} \\ \\ \bold{ \large{ = - \frac{b \sqrt{ {b}^{2} - 4ac} }{ {a}^{2} } }} \\ \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{ \color{blue}{ \mathfrak{ \bold{hope \: it \: helps}}}}$