Let $(a,c)$ be the roots of the equation $x ^ 2 + ax - b = 0$.
Let $(b,d)$ be the roots of the equation $x ^ 2 + cx + d = 0$.
Find all the possible real values for $a, b, c, d$.
NOTE: I have made very little progress towards the answer and any partial answers/hints/explanations/ answers would be extremely helpful. Thanks in advance. :)
HINT:
Immediately we have $a+c=-a\iff c=-2a$ and $bd=d\iff d(b-1)=0$
and $ca=-b\implies(-2a)a=-b\implies b=2a^2$
and $b+d=-c\implies2a^2+d=-2a\iff d=-2a-2a^2$
If $d=0,-2a-2a^2=0$
If $b-1=0\iff b=1,2a^2=1$