Solve m^2+12m-30<0

$$m ^ { 2 } + 12 m - 30 < 0$$

Solve for m

$$m\in \left(-\sqrt{66}-6,\sqrt{66}-6\right)$$

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.

$$m^{2}+12m-30=0$$

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, 12 for b, and -30 for c in the quadratic formula.

$$m=\frac{-12±\sqrt{12^{2}-4\times 1\left(-30\right)}}{2}$$

Do the calculations.

$$m=\frac{-12±2\sqrt{66}}{2}$$

Solve the equation m=\frac{-12±2\sqrt{66}}{2} when ± is plus and when ± is minus.

$$m=\sqrt{66}-6 m=-\sqrt{66}-6$$

Rewrite the inequality by using the obtained solutions.

$$\left(m-\left(\sqrt{66}-6\right)\right)\left(m-\left(-\sqrt{66}-6\right)\right)<0$$

For the product to be negative, m-\left(\sqrt{66}-6\right) and m-\left(-\sqrt{66}-6\right) have to be of the opposite signs. Consider the case when m-\left(\sqrt{66}-6\right) is positive and m-\left(-\sqrt{66}-6\right) is negative.

$$m-\left(\sqrt{66}-6\right)>0 m-\left(-\sqrt{66}-6\right)<0$$

This is false for any m.

$$m\in \emptyset$$

Consider the case when m-\left(-\sqrt{66}-6\right) is positive and m-\left(\sqrt{66}-6\right) is negative.

$$m-\left(-\sqrt{66}-6\right)>0 m-\left(\sqrt{66}-6\right)<0$$

The solution satisfying both inequalities is m\in \left(-\left(\sqrt{66}+6\right),\sqrt{66}-6\right).

$$m\in \left(-\left(\sqrt{66}+6\right),\sqrt{66}-6\right)$$

The final solution is the union of the obtained solutions.

$$m\in \left(-\sqrt{66}-6,\sqrt{66}-6\right)$$