Solve x^2-3x-3leq0

$$x ^ { 2 } - 3 x - 3 \leq 0$$

Solve for x

$$x\in \begin{bmatrix}\frac{3-\sqrt{21}}{2},\frac{\sqrt{21}+3}{2}\end{bmatrix}$$

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.

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

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

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

Do the calculations.

$$x=\frac{3±\sqrt{21}}{2}$$

Solve the equation x=\frac{3±\sqrt{21}}{2} when ± is plus and when ± is minus.

$$x=\frac{\sqrt{21}+3}{2} x=\frac{3-\sqrt{21}}{2}$$

Rewrite the inequality by using the obtained solutions.

$$\left(x-\frac{\sqrt{21}+3}{2}\right)\left(x-\frac{3-\sqrt{21}}{2}\right)\leq 0$$

For the product to be ≤0, one of the values x-\frac{\sqrt{21}+3}{2} and x-\frac{3-\sqrt{21}}{2} has to be ≥0 and the other has to be ≤0. Consider the case when x-\frac{\sqrt{21}+3}{2}\geq 0 and x-\frac{3-\sqrt{21}}{2}\leq 0.

$$x-\frac{\sqrt{21}+3}{2}\geq 0 x-\frac{3-\sqrt{21}}{2}\leq 0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when x-\frac{\sqrt{21}+3}{2}\leq 0 and x-\frac{3-\sqrt{21}}{2}\geq 0.

$$x-\frac{3-\sqrt{21}}{2}\geq 0 x-\frac{\sqrt{21}+3}{2}\leq 0$$

The solution satisfying both inequalities is x\in \left[\frac{3-\sqrt{21}}{2},\frac{\sqrt{21}+3}{2}\right].

$$x\in \begin{bmatrix}\frac{3-\sqrt{21}}{2},\frac{\sqrt{21}+3}{2}\end{bmatrix}$$

The final solution is the union of the obtained solutions.

$$x\in \begin{bmatrix}\frac{3-\sqrt{21}}{2},\frac{\sqrt{21}+3}{2}\end{bmatrix}$$