Solve 2x^2-2x-2

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

Factor

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

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.

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

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.

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

Square -2.

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

Multiply -4 times 2.

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

Multiply -8 times -2.

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

Add 4 to 16.

$$x=\frac{-\left(-2\right)±\sqrt{20}}{2\times 2}$$

Take the square root of 20.

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

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{5}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{2±2\sqrt{5}}{4}$$

Now solve the equation x=\frac{2±2\sqrt{5}}{4} when ± is plus. Add 2 to 2\sqrt{5}.

$$x=\frac{2\sqrt{5}+2}{4}$$

Divide 2+2\sqrt{5} by 4.

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

Now solve the equation x=\frac{2±2\sqrt{5}}{4} when ± is minus. Subtract 2\sqrt{5} from 2.

$$x=\frac{2-2\sqrt{5}}{4}$$

Divide 2-2\sqrt{5} by 4.

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

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}.

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

Evaluate

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