Solve a-6+a^2<0

$$a - 6 + a ^ { 2 } < 0$$

Solve for a

$$a\in \left(-3,2\right)$$

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.

$$a-6+a^{2}=0$$

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, 1 for b, and -6 for c in the quadratic formula.

$$a=\frac{-1±\sqrt{1^{2}-4\times 1\left(-6\right)}}{2}$$

Do the calculations.

$$a=\frac{-1±5}{2}$$

Solve the equation a=\frac{-1±5}{2} when ± is plus and when ± is minus.

$$a=2 a=-3$$

Rewrite the inequality by using the obtained solutions.

$$\left(a-2\right)\left(a+3\right)<0$$

For the product to be negative, a-2 and a+3 have to be of the opposite signs. Consider the case when a-2 is positive and a+3 is negative.

$$a-2>0 a+3<0$$

This is false for any a.

$$a\in \emptyset$$

Consider the case when a+3 is positive and a-2 is negative.

$$a+3>0 a-2<0$$

The solution satisfying both inequalities is a\in \left(-3,2\right).

$$a\in \left(-3,2\right)$$

The final solution is the union of the obtained solutions.

$$a\in \left(-3,2\right)$$