An algebra problem by Bala vidyadharan

We can use the quadratic formula to find solutions for the given equation. They are: $\frac{-2+\sqrt{4-12p(p-1)}}{6} \text{and} \frac{-2-\sqrt{4-12p(p-1)}}{6}.$ The second of these cannot be positive, so we are looking for p values that make the first expression positive. This occurs when $\sqrt{4-12p(p-1)}>2$ .

Solving for p:

$\sqrt{4-12p(p-1)}>2,$

$4-12p(p-1)>4,$

$-12p(p-1)>0,$

$p(p-1)<0.$

Since p(p-1) is an upward parabola with x-intercepts at 0 and 1,

$p(p-1)<0$ for $p \in (0,1).$ These are the p values that give us solutions of opposite sign.