Solve 3-5x-2x^2>0

$$3 - 5 x - 2 x ^ { 2 } > 0$$

Solve for x

$$x\in \left(-3,\frac{1}{2}\right)$$

Steps Using the Quadratic Formula

Multiply the inequality by -1 to make the coefficient of the highest power in 3-5x-2x^{2} positive. Since -1 is negative, the inequality direction is changed.

$$-3+5x+2x^{2}<0$$

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.

$$-3+5x+2x^{2}=0$$

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

$$x=\frac{-5±\sqrt{5^{2}-4\times 2\left(-3\right)}}{2\times 2}$$

Do the calculations.

$$x=\frac{-5±7}{4}$$

Solve the equation x=\frac{-5±7}{4} when ± is plus and when ± is minus.

$$x=\frac{1}{2} x=-3$$

Rewrite the inequality by using the obtained solutions.

$$2\left(x-\frac{1}{2}\right)\left(x+3\right)<0$$

For the product to be negative, x-\frac{1}{2} and x+3 have to be of the opposite signs. Consider the case when x-\frac{1}{2} is positive and x+3 is negative.

$$x-\frac{1}{2}>0 x+3<0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when x+3 is positive and x-\frac{1}{2} is negative.

$$x+3>0 x-\frac{1}{2}<0$$

The solution satisfying both inequalities is x\in \left(-3,\frac{1}{2}\right).

$$x\in \left(-3,\frac{1}{2}\right)$$

The final solution is the union of the obtained solutions.

$$x\in \left(-3,\frac{1}{2}\right)$$