Question

if alpha and beta are the zeroes of the polynomial 5x2 - p(x+1) +c such that (alpha+1) (beta+1) = 0 then find c

Answer 1

$( \alpha + 1)( \beta + 1) = 0 \\ ( \alpha \beta )+ (\alpha + \beta )+ 1 = 0...(1)\\ from \: eqn \: \\ 5 {x}^{2} - p(x + 1) + c = 0 \\ or \\5 {x}^{2} - px + ( - p + c) = 0 \\ \alpha + \beta = \frac{p}{5} \\ \alpha \beta = \frac{ - p + c}{5} \\ from \: (1) \\ \alpha \beta + \alpha + \beta + 1 = 0 \\ \frac{ - p + c}{5} + \frac{p}{5} + 1 = 0 \\ \frac{c}{5} + 1 = 0 \\ c = - 5$
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