i need to find out the equation, the roots are given:
-1+2√2 and -1-2√2
2 2
pls help
$\displaystyle x^2-(sum \ of\ roots)x+product \ of \ roots=0$
$\displaystyle \\ \alpha= \frac{-1+2\sqrt{2}}{2}\ \ \beta= \frac{-1-2\sqrt{2}}{2}$
$\displaystyle \alpha+ \beta= \frac{-1+2\sqrt{2}}{2}\ +\frac{-1-2\sqrt{2}}{2}\ = \frac{-1}{2}+\frac{2\sqrt{2}}{2}\ +\frac{-1}{2}-\frac{2\sqrt{2}}{2}$
$\displaystyle \alpha+ \beta=\frac{-2}{2} =\ -1$
$\displaystyle \\ \alpha \beta= \{ \frac{-1+2\sqrt{2}}{2}\ \} \{ \frac{-1-2\sqrt{2}}{2} \} \ = \{ \frac{1-2\sqrt{2}}{2}\ \} \{ \frac{1+2\sqrt{2}}{2} \}$
$\displaystyle \\ \alpha \beta=\frac {\{1^2-(2\sqrt{2})^2\}}{4} \ = \frac{(1-8)}{4}=\frac{-7}{4}$
therefore ,the equation is
$\displaystyle \\ x^2-(\alpha +\beta)x+\alpha. \beta=0$
$\displaystyle \\ x^2-(-1)x+\frac{-7}{4}=0$
$\displaystyle \\ 4x^2+4x-7=0$