Solve x^2-x+1>0

$${ x }^{ 2 } -x+1 > 0$$

Solve for x

$$x\in \mathrm{R}$$

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.

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

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.

$$x=\frac{-\left(-1\right)±\sqrt{\left(-1\right)^{2}-4\times 1\times 1}}{2}$$

Do the calculations.

$$x=\frac{1±\sqrt{-3}}{2}$$

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

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

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

$$x\in \mathrm{R}$$