Solve {c}{2m^2+3m}{+11geq0}

$$\left. \begin{array} { c } { 2 m ^ { 2 } + 3 m } \\ { + 11 \geq 0 } \end{array} \right.$$

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.

$$2m^{2}+3m+11=0$$

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 2 for a, 3 for b, and 11 for c in the quadratic formula.

$$m=\frac{-3±\sqrt{3^{2}-4\times 2\times 11}}{2\times 2}$$

Do the calculations.

$$m=\frac{-3±\sqrt{-79}}{4}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Expression 2m^{2}+3m+11 has the same sign for any m. To determine the sign, calculate the value of the expression for m=0.

$$2\times 0^{2}+3\times 0+11=11$$

The value of the expression 2m^{2}+3m+11 is always positive. Inequality holds for m\in \mathrm{R}.

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