Find all quadruples of real numbers $a$, $b$, $c$, and $d$ such that $a$ and $b$ are distinct roots of quadratic $x^2 + cx + d = 0$, while $c$ and $d$ are the roots of $x^2 + ax + b = 0$.
My initial thought was to simplify the equations and solve as a system, but this did not work. What is the best way to approach problems like these, and how can I solve this?
Your system of equations approach will work here. We have $4$ equations: $$a+b+c=0.....(1)$$ $$ab=d.....(2)$$ $$a+c+d=0.....(3)$$ $$cd=b.....(4)$$ From $(1)$ and $(3)$, we have $b=d. $ Substituting in $(2)$, we get: Either $b=d=0$ or $a=1$. If the first case is true, then $a,c$ can be any real numbers such that $a+c=0$ and $a,c\neq 0$. Otherwise, we get, from $(4)$, $c=1$. So $b=d=-2$. So one quadruplet is $(a,0,-a,0), a\neq 0$ and the other is $(1,-2,1,-2)$.