Solve 16x^2-16x-16

$$16 { x }^{ 2 } -16x-16$$

Factor

$$16\left(x-\frac{1-\sqrt{5}}{2}\right)\left(x-\frac{\sqrt{5}+1}{2}\right)$$

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.

$$16x^{2}-16x-16=0$$

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.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 16\left(-16\right)}}{2\times 16}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 16\left(-16\right)}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-16\right)±\sqrt{256-64\left(-16\right)}}{2\times 16}$$

Multiply -64 times -16.

$$x=\frac{-\left(-16\right)±\sqrt{256+1024}}{2\times 16}$$

Add 256 to 1024.

$$x=\frac{-\left(-16\right)±\sqrt{1280}}{2\times 16}$$

Take the square root of 1280.

$$x=\frac{-\left(-16\right)±16\sqrt{5}}{2\times 16}$$

The opposite of -16 is 16.

$$x=\frac{16±16\sqrt{5}}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{16±16\sqrt{5}}{32}$$

Now solve the equation x=\frac{16±16\sqrt{5}}{32} when ± is plus. Add 16 to 16\sqrt{5}.

$$x=\frac{16\sqrt{5}+16}{32}$$

Divide 16+16\sqrt{5} by 32.

$$x=\frac{\sqrt{5}+1}{2}$$

Now solve the equation x=\frac{16±16\sqrt{5}}{32} when ± is minus. Subtract 16\sqrt{5} from 16.

$$x=\frac{16-16\sqrt{5}}{32}$$

Divide 16-16\sqrt{5} by 32.

$$x=\frac{1-\sqrt{5}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1+\sqrt{5}}{2} for x_{1} and \frac{1-\sqrt{5}}{2} for x_{2}.

$$16x^{2}-16x-16=16\left(x-\frac{\sqrt{5}+1}{2}\right)\left(x-\frac{1-\sqrt{5}}{2}\right)$$

Evaluate

$$16\left(x^{2}-x-1\right)$$