Solve 3+2a-a^2>0

$$3 + 2 a - a ^ { 2 } > 0$$

Solve for a

$$a\in \left(-1,3\right)$$

Steps Using the Quadratic Formula

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

$$-3-2a+a^{2}<0$$

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.

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

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

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

Do the calculations.

$$a=\frac{2±4}{2}$$

Solve the equation a=\frac{2±4}{2} when ± is plus and when ± is minus.

$$a=3 a=-1$$

Rewrite the inequality by using the obtained solutions.

$$\left(a-3\right)\left(a+1\right)<0$$

For the product to be negative, a-3 and a+1 have to be of the opposite signs. Consider the case when a-3 is positive and a+1 is negative.

$$a-3>0 a+1<0$$

This is false for any a.

$$a\in \emptyset$$

Consider the case when a+1 is positive and a-3 is negative.

$$a+1>0 a-3<0$$

The solution satisfying both inequalities is a\in \left(-1,3\right).

$$a\in \left(-1,3\right)$$

The final solution is the union of the obtained solutions.

$$a\in \left(-1,3\right)$$