Solve x^2-22x-117

$${ x }^{ 2 } -22x-117$$

Factor

$$\left(x-\left(11-\sqrt{238}\right)\right)\left(x-\left(\sqrt{238}+11\right)\right)$$

Steps Using the Quadratic Formula

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.

$$x^{2}-22x-117=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\left(-117\right)}}{2}$$

Square -22.

$$x=\frac{-\left(-22\right)±\sqrt{484-4\left(-117\right)}}{2}$$

Multiply -4 times -117.

$$x=\frac{-\left(-22\right)±\sqrt{484+468}}{2}$$

Add 484 to 468.

$$x=\frac{-\left(-22\right)±\sqrt{952}}{2}$$

Take the square root of 952.

$$x=\frac{-\left(-22\right)±2\sqrt{238}}{2}$$

The opposite of -22 is 22.

$$x=\frac{22±2\sqrt{238}}{2}$$

Now solve the equation x=\frac{22±2\sqrt{238}}{2} when ± is plus. Add 22 to 2\sqrt{238}.

$$x=\frac{2\sqrt{238}+22}{2}$$

Divide 22+2\sqrt{238} by 2.

$$x=\sqrt{238}+11$$

Now solve the equation x=\frac{22±2\sqrt{238}}{2} when ± is minus. Subtract 2\sqrt{238} from 22.

$$x=\frac{22-2\sqrt{238}}{2}$$

Divide 22-2\sqrt{238} by 2.

$$x=11-\sqrt{238}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 11+\sqrt{238} for x_{1} and 11-\sqrt{238} for x_{2}.

$$x^{2}-22x-117=\left(x-\left(\sqrt{238}+11\right)\right)\left(x-\left(11-\sqrt{238}\right)\right)$$

Evaluate

$$x^{2}-22x-117$$