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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.

MathDona
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  • (i) is what says the quadratic has distinct real roots – Will Jagy Oct 24 '22 at 01:34
  • If (i) failed, then the equation have only one root or no root. If there is no root, $x=p$ lies within the roots cannot happen. – Abel Wong Oct 24 '22 at 01:40
  • Which statements are conditions for which here? Usually a sentence beginning with "If $x = p$ lies within the roots of a quadratic equation" would end by stating one or more other facts that are implied by $x=p$ lying between the roots of the equation. None of those other facts would be superfluous or sufficient; they're all merely consequences of a hypothetical fact already given. – David K Oct 24 '22 at 02:07

1 Answers1

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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 $A\cdot f(p)\gt0$.

Constantinos Pisimisis
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