A sequence of quadratic equations

Let $p\le q$ be roots of the (real) quadratic equation $x^2+ax+b=0$, $|p|+|q|\ne 0.$ Form the new equation $x^2+px+q=0$, find its real roots (if exist), etc. For example, if $a=3, b=2$, then $p=-2,q=-1$, $x^2-2x-1=0$ has two real roots $p_1=(2-\sqrt{2})/2$ and $q_1=(2+\sqrt{2})/2$ (note that $p_1\le q_1$) but the equation $x^2+p_1x+q_1$ does not have real roots, the process ends.

Question: What is the longest possible sequence of quadratic equations we can get?

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