Solve -2x^2-3x-6leq0

$$-2 { x }^{ 2 } -3x-6 \leq 0$$

Solve for x

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

Steps Using the Quadratic Formula

Multiply the inequality by -1 to make the coefficient of the highest power in -2x^{2}-3x-6 positive. Since -1 is negative, the inequality direction is changed.

$$2x^{2}+3x+6\geq 0$$

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.

$$2x^{2}+3x+6=0$$

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 6 for c in the quadratic formula.

$$x=\frac{-3±\sqrt{3^{2}-4\times 2\times 6}}{2\times 2}$$

Do the calculations.

$$x=\frac{-3±\sqrt{-39}}{4}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Expression 2x^{2}+3x+6 has the same sign for any x. To determine the sign, calculate the value of the expression for x=0.

$$2\times 0^{2}+3\times 0+6=6$$

The value of the expression 2x^{2}+3x+6 is always positive. Inequality holds for x\in \mathrm{R}.

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