Solve x^2+5x+6leq0

$$x ^ { 2 } + 5 x + 6 \leq 0$$

Solve for x

$$x\in \begin{bmatrix}-3,-2\end{bmatrix}$$

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.

$$x^{2}+5x+6=0$$

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, 5 for b, and 6 for c in the quadratic formula.

$$x=\frac{-5±\sqrt{5^{2}-4\times 1\times 6}}{2}$$

Do the calculations.

$$x=\frac{-5±1}{2}$$

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

$$x=-2 x=-3$$

Rewrite the inequality by using the obtained solutions.

$$\left(x+2\right)\left(x+3\right)\leq 0$$

For the product to be ≤0, one of the values x+2 and x+3 has to be ≥0 and the other has to be ≤0. Consider the case when x+2\geq 0 and x+3\leq 0.

$$x+2\geq 0 x+3\leq 0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when x+2\leq 0 and x+3\geq 0.

$$x+3\geq 0 x+2\leq 0$$

The solution satisfying both inequalities is x\in \left[-3,-2\right].

$$x\in \begin{bmatrix}-3,-2\end{bmatrix}$$

The final solution is the union of the obtained solutions.

$$x\in \begin{bmatrix}-3,-2\end{bmatrix}$$