Solve 4x-2x^2+2

$$4 x - 2 x ^ { 2 } + 2$$

Factor

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

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.

$$-2x^{2}+4x+2=0$$

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.

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

Square 4.

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

Multiply -4 times -2.

$$x=\frac{-4±\sqrt{16+8\times 2}}{2\left(-2\right)}$$

Multiply 8 times 2.

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

Add 16 to 16.

$$x=\frac{-4±\sqrt{32}}{2\left(-2\right)}$$

Take the square root of 32.

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

Multiply 2 times -2.

$$x=\frac{-4±4\sqrt{2}}{-4}$$

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

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

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

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

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

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

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

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

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

$$-2x^{2}+4x+2=-2\left(x-\left(1-\sqrt{2}\right)\right)\left(x-\left(\sqrt{2}+1\right)\right)$$

Evaluate

$$2+4x-2x^{2}$$