Question

find the quadratic polynomial of whose zeroes are 2+underoot 5 and sum of zeroes is 4​

Answer 1

Answer:

$one \: zero \: \alpha = 2 + \sqrt{5} \\ sum \: of \: zeroes \: \alpha + \beta \: = 4 \\ 2 + \sqrt{5} + \beta = 4 \\ \beta \: = 4 - 2 - \sqrt{5} = 2 - \sqrt{5} \\ product \: of \: zeroes \: \alpha \beta = (2 + \sqrt{5} )(2 - \sqrt{5} ) = 4 - 5 = - 1$

$so \: required \: quadratic \: polynomial \: is \: \\ f(x) = {x}^{2} - ( \alpha + \beta )x + \alpha \beta \\ = {x}^{2} - 4x - 1$