parmenides51 wrote: Laura and Daniel play with quadratic polynomials. First Laura says a nonzero real number $r$. Then Daniel says a nonzero real number $s$, and then again Laura says another nonzero real number $t$. Finally. Daniel writes the polynomial $P(x) = ax^2 + bx + c$ where $a,b$, and $c$ are $r,s$, and $t$ in some order Daniel chooses. Laura wins if the equation $P(x) = 0$ has two different real solutions, and Daniel wins otherwise. Determine who has a winning strategy and describe that strategy. Laura has a winning strategy : Laura choose any $r>0$ If Daniel chooses $s<0$, Laura chooses any nonzero $t\in\left(\frac {r^2}{4s},\frac {s^2}{4r}\right)$ (which is not empty since it contains $0$) If Daniel chooses $s>0$, Laura chooses any $t<-2\sqrt{rs}$