Solve x^2-10x-6>0

$${ x }^{ 2 } -10x-6 > 0$$

Solve for x

$$x\in \left(-\infty,5-\sqrt{31}\right)\cup \left(\sqrt{31}+5,\infty\right)$$

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.

$$x^{2}-10x-6=0$$

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, -10 for b, and -6 for c in the quadratic formula.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 1\left(-6\right)}}{2}$$

Do the calculations.

$$x=\frac{10±2\sqrt{31}}{2}$$

Solve the equation x=\frac{10±2\sqrt{31}}{2} when ± is plus and when ± is minus.

$$x=\sqrt{31}+5 x=5-\sqrt{31}$$

Rewrite the inequality by using the obtained solutions.

$$\left(x-\left(\sqrt{31}+5\right)\right)\left(x-\left(5-\sqrt{31}\right)\right)>0$$

For the product to be positive, x-\left(\sqrt{31}+5\right) and x-\left(5-\sqrt{31}\right) have to be both negative or both positive. Consider the case when x-\left(\sqrt{31}+5\right) and x-\left(5-\sqrt{31}\right) are both negative.

$$x-\left(\sqrt{31}+5\right)<0 x-\left(5-\sqrt{31}\right)<0$$

The solution satisfying both inequalities is x<5-\sqrt{31}.

$$x<5-\sqrt{31}$$

Consider the case when x-\left(\sqrt{31}+5\right) and x-\left(5-\sqrt{31}\right) are both positive.

$$x-\left(5-\sqrt{31}\right)>0 x-\left(\sqrt{31}+5\right)>0$$

The solution satisfying both inequalities is x>\sqrt{31}+5.

$$x>\sqrt{31}+5$$

The final solution is the union of the obtained solutions.

$$x<5-\sqrt{31}\text{; }x>\sqrt{31}+5$$