Solve -3x^2-11x-10>0

$$- 3 x ^ { 2 } - 11 x - 10 > 0$$

Solve for x

$$x\in \left(-2,-\frac{5}{3}\right)$$

Steps Using the Quadratic Formula

Multiply the inequality by -1 to make the coefficient of the highest power in -3x^{2}-11x-10 positive. Since -1 is negative, the inequality direction is changed.

$$3x^{2}+11x+10<0$$

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.

$$3x^{2}+11x+10=0$$

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 3 for a, 11 for b, and 10 for c in the quadratic formula.

$$x=\frac{-11±\sqrt{11^{2}-4\times 3\times 10}}{2\times 3}$$

Do the calculations.

$$x=\frac{-11±1}{6}$$

Solve the equation x=\frac{-11±1}{6} when ± is plus and when ± is minus.

$$x=-\frac{5}{3} x=-2$$

Rewrite the inequality by using the obtained solutions.

$$3\left(x+\frac{5}{3}\right)\left(x+2\right)<0$$

For the product to be negative, x+\frac{5}{3} and x+2 have to be of the opposite signs. Consider the case when x+\frac{5}{3} is positive and x+2 is negative.

$$x+\frac{5}{3}>0 x+2<0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when x+2 is positive and x+\frac{5}{3} is negative.

$$x+2>0 x+\frac{5}{3}<0$$

The solution satisfying both inequalities is x\in \left(-2,-\frac{5}{3}\right).

$$x\in \left(-2,-\frac{5}{3}\right)$$

The final solution is the union of the obtained solutions.

$$x\in \left(-2,-\frac{5}{3}\right)$$