Question

$if \alpha and \beta are the zeroes of the polynomial f(x)=x^{2} -5x+k such that \alpha -\beta =1,find the value of k$

Answer 1

$\alpha - \beta = 1 - - - (1)\\ according \: to \: quadratic \: equation \: \\ \alpha + \beta = \frac{ - b}{a} \\ \alpha + \beta = 5 - - - (2) \\ \\ \alpha \beta = \frac{c}{a} \\ \alpha \beta = k - - (3) \\ \\ adding \: eq.(1)and \: (2) \\ 2 \alpha = 6 \\ \alpha = 3 \\ \\ put \: value \: of \: \alpha \: in \: eq.2 \\ \beta = 5 - \alpha \\ so \: \beta = 5 - 3 \\ \beta = 2 \\ \\ k = \alpha \beta \\ so \\ \\ k = 3 \times 2 \\ \\ k = 6 \: ans.$