1. ## help

Prove or Disprove that there exists a real number x such that x^2+x+1 is less than or equal to zero.

2. Do you know the intermediate value theorem?

3. yes

4. Why not just use the quadratic equation?

$\displaystyle x = \frac {-b\pm \sqrt{b^2 - 4ac}}{2a}$

5. Yes that gives me a solution but then how do i prove that the equation works? Thats where i am now stuck.

6. From the quadratic equation, you know there are no real roots, meaning the function is never 0 (either always negative, or always positive). With that, you can note that letting x=0 gives you a value of 1, so the function is always positive.

Not quite the most rigorous proof and probably not what your professor is looking for, but it is valid.

Edit: Also, I forgot to add that this works because the function is continuous over all real numbers.

7. Originally Posted by natewalker205
Yes that gives me a solution but then how do i prove that the equation works? Thats where i am now stuck.
Well the quadratic formula tells you where you have a value of x such that
$\displaystyle x^2 + x + 1 = 0$
which is one of the criterion of your proof. So the existence of x proves that the function can be less than or equal to 0.

Likely the best method is the intermediate value theorem, if you can do it that way.

-Dan