Solve n^2+n+1>0

$$n ^ { 2 } + n + 1 > 0$$

Solve for n

$$n\in \mathrm{R}$$

Steps Using the Quadratic Formula

To solve the inequality, factor the left hand side. Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$n^{2}+n+1=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, 1 for b, and 1 for c in the quadratic formula.

$$n=\frac{-1±\sqrt{1^{2}-4\times 1\times 1}}{2}$$

Do the calculations.

$$n=\frac{-1±\sqrt{-3}}{2}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Expression n^{2}+n+1 has the same sign for any n. To determine the sign, calculate the value of the expression for n=0.

$$0^{2}+0+1=1$$

The value of the expression n^{2}+n+1 is always positive. Inequality holds for n\in \mathrm{R}.

$$n\in \mathrm{R}$$