Question

two roots of quadratic equations are given frame the equation ​

Answer 1

$Let \: \alpha \: and \: \beta \: be \: the \: roots \: of \: the \: quadratic \: equation \\ \\Let \: \alpha = 1 - 3 \sqrt{5} \: and \: \beta = 1 + 3 \sqrt{5} \\ \\ \alpha + \beta = 1 - 3 \sqrt{5} + 1 + 3 \sqrt{5} \\ = 2 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \alpha \times \beta = (1 - 3 \sqrt{5}) \times (1 + 3 \sqrt{5} ) \\ = {1}^{2} - (3 \sqrt{5})^{2} \: \: \: \: \: \: \: \: \: \\ = 1 - 9 \times 5 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ = 1 - 45 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ = 44 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

Then the required quadratic equation is

${x}^{2} - ( \alpha + \beta ) \: x \: + \: \alpha \beta \: = 0 \\ {x}^{2} \: - \: 2 \: x \: - 44 = 0$

{x}^{2} - 2 x - 44 = 0 is the equation formed.