Question

if alpha and beta are zeroes of quadratic polynomial f(x)=ax²+bx+c,then evaluate alpha - beta

Answer 1

$\alpha \: and \: \beta \: are \: zeroes \: of \: f(x) = a {x}^{2} + bx + c \\ we \: know \: that \: zeroes \: of \: a \: quadratic \: equation \: are \: \frac{ - b + \sqrt{ {b}^{2} - 4ac } }{2a} or \: \frac{ - b - \sqrt{ {b}^{2} - 4ac } }{2a} \\ i.e. \alpha = \frac{ - b + \sqrt{ {b}^{2} - 4ac } }{2a} \\ \\ \beta = \frac{ - b - \sqrt{ {b}^{2} - 4ac } }{2a} \\ \\ \alpha - \beta = \frac{ - b + \sqrt{ {b}^{2} - 4ac } }{2a} - \frac{ - b - \sqrt{ {b}^{2} - 4ac } }{2a} \\ = \frac{ - b + \sqrt{ {b}^{2} - 4ac } + b + \sqrt{ {b}^{2} - 4ac }}{2a} \\ = \frac{2 \sqrt{ {b}^{2} - 4ac }}{2a} \\ \\ = \frac{ \sqrt{ {b}^{2} - 4ac }}{a}$