# If $a$, $b$, and $c$ are reals for which there is a real number $x \not = 0$ such that $ax^2 + bx + c = 0$, then $cx^2 + bx + a$ has a rational root.

I have the following proposition:

If $a$, $b$, and $c$ are real numbers for which there is a real number $x \not = 0$ such that $ax^2 + bx + c = 0$, then $cx^2 + bx + a$ has a rational root.

Therefore, the hypothesis and conclusion are as follows:

Hypothesis: $a$, $b$, and $c$ are real numbers for which there is a real number $x \not = 0$ such that $ax^2 + bx + c = 0$.

Conclusion: $cx^2 + bx + a$ has a rational root.

I'm wondering if it is completely equivalent to reword the hypothesis as follows:

Hypothesis: For all real numbers $a$, $b$, and $c$, there is a real number $x \not = 0$ such that $ax^2 + bx + c = 0$.

I would greatly appreciate it if someone could please take the time to clarify my thoughts.

• The proposition is false. A completely equivalent rewording could be $0=1$. – dxiv Feb 2 '17 at 4:36
• Just looking at the hypothesis, no, the rewording is not equivalent. The "rewording" is a statement that is always false. – Jonas Meyer Feb 2 '17 at 4:38
• @JonasMeyer Ahh, I see. Is it because for some $a$, $b$, and $c$ we would have complex roots? – The Pointer Feb 2 '17 at 4:40
• @ThePointer: Yes. $x^2+1$ shows the second hypothesis is a false statement. $x^2-2$ shows the proposition is false. – Jonas Meyer Feb 2 '17 at 4:41
• @JonasMeyer I understand. Thank you all very much for the assistance. :) – The Pointer Feb 2 '17 at 4:42