Originally Posted by
mathaddict HI
For the first problem ,
$\displaystyle x^2+ax-b=0$ its roots are a and c
$\displaystyle a+c=-a$ ---1
$\displaystyle ac=-b$ ----2
$\displaystyle x^2+cx+d=0$ its roots are b and d ,
$\displaystyle b+d=-c$ --- 3
$\displaystyle bd=d$ ---4
From 4 , $\displaystyle bd=d\Rightarrow b=1$
From 1 , $\displaystyle 2a=-c $
At 2 , $\displaystyle a(-2a)=-b$
$\displaystyle 2a^2=b$
$\displaystyle 2a^2=1$
$\displaystyle a=\pm\frac{1}{\sqrt{2}}$
At 1 , $\displaystyle 2(\pm\frac{1}{\sqrt{2}})=-c$
$\displaystyle c=2(\mp\frac{1}{\sqrt{2}})$
$\displaystyle c=\mp\sqrt{2}$
Then , from 2 ,
$\displaystyle 1+d=-c$
$\displaystyle 1+d=\pm\sqrt{2}$
$\displaystyle
d=-1\pm\sqrt{2}
$