3

I have been working on a math problem involving inequalities and have come across a question that has left me puzzled. I would greatly appreciate some insight and clarification on the correct answer choice.

The question states:

The inequality $ax^2 + bx + c > 0$ (where $a$, $b$, and $c$ are real numbers) has finite positive integer solutions. Based on this information, which conclusion can be drawn?"

The answer choices are:

  • (a) $a > 0$
  • (b) $a \neq 0$
  • (c) $a < 0$
  • (d) None of the above

I am particularly confused about why the correct answer is (d) - None of the above. It seems counterintuitive that we cannot deduce any information about the value of $a$ based on the given condition. Could someone kindly explain the reasoning behind this? What additional information do we need to draw a definitive conclusion? Is there any particular concept or property that I am missing here?

Digitallis
  • 3,780
  • 1
  • 9
  • 31
Bishop_1
  • 359
  • Not sure what "finite positive integer solutions " means. Is a single solution good enough? – lulu Jun 10 '23 at 13:01
  • @lulu How can we determine that a is not a positive number? Does a=0 satisfy this condition? If a=0, it leads to a negative solution, specifically x=-c/b. When a is both greater than 0 and less than 0, we obtain two distinct real roots with opposite signs, and potentially even with the same sign. – Bishop_1 Jun 10 '23 at 15:31
  • Not following. If $(a,b,c)=(0,1,-1)$ then we get the equation $x-1=0$ which has $x=1$ as a positive, integer solution. – lulu Jun 10 '23 at 15:43
  • And, for completeness, if $(a,b,c)=(-1,3,-2)$ then we get $-x^2+3x-2=0$ which has solutions $x=1,2$. And if $(a,b,c)=(1,-3,2)$ then we get $x^2-3x+2=0$ which again has solutions $x=1,2$. So we can have $a=0$, we can have $a<0$, and we can have $a>0$, as desired. – lulu Jun 10 '23 at 15:46
  • @lulu But the question is about the inequality $ax^2+bx+c > 0$. I think the conclusion would be $a\le 0$, which is "(d) None of the above". – peterwhy Jun 10 '23 at 22:01
  • @peterwhy you're right, I took it as just asking about the roots. – lulu Jun 10 '23 at 23:28

1 Answers1

0

There is no way to deduce any information of $a$!

Letting $a>0, b=c=0$ or $a=0, b=c=1$ or $a<0, b=0, c=1$ will all give you finite positive integer solutions.

IraeVid
  • 3,216
  • How do we know that a is not positive and a=0 fulfills that? a=0 gives a negative solution, ie x=-c/b? When a>0 and a<0 we get two different real roots with different signs and maybe even with the same sign. – Bishop_1 Jun 10 '23 at 15:30
  • 1
    But the question is about the inequality $ax^2+bx+c>0$. For two of your cases ($a>0, b=c=0$, or $a=0,b=c=1$) there are infinite positive integer solutions of $x$. Sorry that I downvoted for this reason. – peterwhy Jun 11 '23 at 00:27