Solve m^2-3m+3geq0

$$m ^ { 2 } - 3 m + 3 \geq 0$$

Solve for m

$$m\in \mathrm{R}$$

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.

$$m^{2}-3m+3=0$$

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, -3 for b, and 3 for c in the quadratic formula.

$$m=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 1\times 3}}{2}$$

Do the calculations.

$$m=\frac{3±\sqrt{-3}}{2}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Expression m^{2}-3m+3 has the same sign for any m. To determine the sign, calculate the value of the expression for m=0.

$$0^{2}-3\times 0+3=3$$

The value of the expression m^{2}-3m+3 is always positive. Inequality holds for m\in \mathrm{R}.

$$m\in \mathrm{R}$$