A condition for the roots of a quadratic equation

Question

If $x=p$ lies within the roots of the quadratic equation $f(x)=Ax^2+Bx+C=0$ then we demand $(i):B^2>4AC$ and $(ii):Af(p)<0$. I want to know if the condition (i) is superfluous here. Or whether the condition $(ii)$ would alone be sufficient here.

Answer 1

For your first demand:

We demand: $(i)\text{ } B^2\gt 4AC$

Since you are saying : if $x=p$ lies within the roots of the quadratic equation it's obvious that there are exactly $2$ roots.So the demand:$B^2\gt 4AC$ it's sufficient and neccesary to prove that you have exactly $2 $ roots.If this demand fails you will have no more than $1$ root.So this demand it's not superfluous .

For your second demand:

$(ii) A\cdot f(p)<0$

Suppose that we have $2$ roots.$\text{ }x_1, x_2$ are the roots of our equation.
Indeed,in the range: $[x_1,x_2]$ we know that: $f(x)\le 0$ but when it comes $A\cdot f(p)$ we don't know anything about $A$
So we distinguish the following cases:
If$\text{ }A\gt 0$,your demand it's correct and sufficient(and by that i mean ,we know that there must be exactly $2$ roots.)
If $\text{ }A\lt0 $,your demand it's wrong because $A\cdot f(p)\gt0$.