OMCC 2001 Problem 5

Let $a,b$ and $c$ real numbers such that the equation $ax^2+bx+c=0$ has two distinct real solutions $p_1,p_2$ and the equation $cx^2+bx+a=0$ has two distinct real solutions $q_1,q_2$ . We know that the numbers $p_1,q_1,p_2,q_2$ in that order, form an arithmetic progression. Show that $a+c=0$ .