## mathmath333 one year ago Find the range of $$x$$

1. mathmath333

\large \color{black}{\begin{align} x^2+x+1>0,\quad x\in \mathbb{R} \hspace{.33em}\\~\\ \end{align}}

2. ParthKohli

$x \in R$

3. ParthKohli

This quadratic expression has no roots, and it points upwards. It means that it must lie above the x-axis always, thus meaning that the inequality holds for all $$x$$.

4. ParthKohli

Are you in 11th?

5. mathmath333

it has imaginary roots

6. ParthKohli

Well, that's obvious. All polynomials have roots. It comes without saying that I mean real roots.

7. ParthKohli

Anyway, do you see why that holds for all $$x$$?

8. mathmath333

cuz $$x\in \mathbb{R},x\cancel{\in} \mathbb{I}$$

9. ParthKohli

In inequalities, we don't talk about complex numbers anyway. Do you see why $$x^2 + x + 1$$ is positive for all $$x$$?

10. ParthKohli

Another way to think is that the minimum value of the function is $$3/4$$, so the function will always be greater than 0.

11. mathmath333

how u got 3/4

12. ParthKohli

$x^2 + x + 1 = \left(x + \frac{1}{2}\right) ^2 + \frac{3}{4}$