Solve 6x^2-8x-1geq0

$$6 x ^ { 2 } - 8 x - 1 \geq 0$$

Solve for x

$$x\in (-\infty,-\frac{\sqrt{22}}{6}+\frac{2}{3}]\cup [\frac{\sqrt{22}}{6}+\frac{2}{3},\infty)$$

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.

$$6x^{2}-8x-1=0$$

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 6 for a, -8 for b, and -1 for c in the quadratic formula.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 6\left(-1\right)}}{2\times 6}$$

Do the calculations.

$$x=\frac{8±2\sqrt{22}}{12}$$

Solve the equation x=\frac{8±2\sqrt{22}}{12} when ± is plus and when ± is minus.

$$x=\frac{\sqrt{22}}{6}+\frac{2}{3} x=-\frac{\sqrt{22}}{6}+\frac{2}{3}$$

Rewrite the inequality by using the obtained solutions.

$$6\left(x-\left(\frac{\sqrt{22}}{6}+\frac{2}{3}\right)\right)\left(x-\left(-\frac{\sqrt{22}}{6}+\frac{2}{3}\right)\right)\geq 0$$

For the product to be ≥0, x-\left(\frac{\sqrt{22}}{6}+\frac{2}{3}\right) and x-\left(-\frac{\sqrt{22}}{6}+\frac{2}{3}\right) have to be both ≤0 or both ≥0. Consider the case when x-\left(\frac{\sqrt{22}}{6}+\frac{2}{3}\right) and x-\left(-\frac{\sqrt{22}}{6}+\frac{2}{3}\right) are both ≤0.

$$x-\left(\frac{\sqrt{22}}{6}+\frac{2}{3}\right)\leq 0 x-\left(-\frac{\sqrt{22}}{6}+\frac{2}{3}\right)\leq 0$$

The solution satisfying both inequalities is x\leq -\frac{\sqrt{22}}{6}+\frac{2}{3}.

$$x\leq -\frac{\sqrt{22}}{6}+\frac{2}{3}$$

Consider the case when x-\left(\frac{\sqrt{22}}{6}+\frac{2}{3}\right) and x-\left(-\frac{\sqrt{22}}{6}+\frac{2}{3}\right) are both ≥0.

$$x-\left(-\frac{\sqrt{22}}{6}+\frac{2}{3}\right)\geq 0 x-\left(\frac{\sqrt{22}}{6}+\frac{2}{3}\right)\geq 0$$

The solution satisfying both inequalities is x\geq \frac{\sqrt{22}}{6}+\frac{2}{3}.

$$x\geq \frac{\sqrt{22}}{6}+\frac{2}{3}$$

The final solution is the union of the obtained solutions.

$$x\leq -\frac{\sqrt{22}}{6}+\frac{2}{3}\text{; }x\geq \frac{\sqrt{22}}{6}+\frac{2}{3}$$