Question

If Alpha and beta are the zeroes of the quadratic polynomial x square - 5x+6. Find the value of 1/alpha + 1/beta - 2alpha * beta?

Answer 1

$one \: zero = \alpha \\ other = \beta \\ \\ now \\ p(x) = {x}^{2} - 5x + 6 \\ \therefore \: \alpha + \beta = \frac{ - b}{a} = 5...(1) \\ and \\ \alpha \times \beta = \frac{c}{a} = 6 ...(2)\\ \\ now \\ \\ \frac{1 }{ \alpha } + \frac{1}{ \beta } - 2 \alpha \beta \\ \\ = \frac{ \alpha + \beta }{ \alpha \beta } - 2 \alpha \beta \\ \\ = \frac{5}{6} - 2(6) \: \: (from \: 1 \: and \: 2) \\ \\ = \frac{5}{6} - 12 \\ \\ = \frac{10 - 12}{12} \\ \\ = - \frac{ \cancel2}{ \cancel12} \\ \\ = - \frac{1}{6} ..........(answer)$