Solve x^2-24x-128

$${ x }^{ 2 } -24x-128$$

Factor

$$\left(x-\left(12-4\sqrt{17}\right)\right)\left(x-\left(4\sqrt{17}+12\right)\right)$$

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.

$$x^{2}-24x-128=0$$

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.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\left(-128\right)}}{2}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\left(-128\right)}}{2}$$

Multiply -4 times -128.

$$x=\frac{-\left(-24\right)±\sqrt{576+512}}{2}$$

Add 576 to 512.

$$x=\frac{-\left(-24\right)±\sqrt{1088}}{2}$$

Take the square root of 1088.

$$x=\frac{-\left(-24\right)±8\sqrt{17}}{2}$$

The opposite of -24 is 24.

$$x=\frac{24±8\sqrt{17}}{2}$$

Now solve the equation x=\frac{24±8\sqrt{17}}{2} when ± is plus. Add 24 to 8\sqrt{17}.

$$x=\frac{8\sqrt{17}+24}{2}$$

Divide 24+8\sqrt{17} by 2.

$$x=4\sqrt{17}+12$$

Now solve the equation x=\frac{24±8\sqrt{17}}{2} when ± is minus. Subtract 8\sqrt{17} from 24.

$$x=\frac{24-8\sqrt{17}}{2}$$

Divide 24-8\sqrt{17} by 2.

$$x=12-4\sqrt{17}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 12+4\sqrt{17} for x_{1} and 12-4\sqrt{17} for x_{2}.

$$x^{2}-24x-128=\left(x-\left(4\sqrt{17}+12\right)\right)\left(x-\left(12-4\sqrt{17}\right)\right)$$

Evaluate

$$x^{2}-24x-128$$