Solve -2x^2+4x+1

$$-2 { x }^{ 2 } +4x+1$$

Factor

$$-2\left(x-\left(-\frac{\sqrt{6}}{2}+1\right)\right)\left(x-\left(\frac{\sqrt{6}}{2}+1\right)\right)$$

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.

$$-2x^{2}+4x+1=0$$

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.

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

Square 4.

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

Multiply -4 times -2.

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

Add 16 to 8.

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

Take the square root of 24.

$$x=\frac{-4±2\sqrt{6}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-4±2\sqrt{6}}{-4}$$

Now solve the equation x=\frac{-4±2\sqrt{6}}{-4} when ± is plus. Add -4 to 2\sqrt{6}.

$$x=\frac{2\sqrt{6}-4}{-4}$$

Divide -4+2\sqrt{6} by -4.

$$x=-\frac{\sqrt{6}}{2}+1$$

Now solve the equation x=\frac{-4±2\sqrt{6}}{-4} when ± is minus. Subtract 2\sqrt{6} from -4.

$$x=\frac{-2\sqrt{6}-4}{-4}$$

Divide -4-2\sqrt{6} by -4.

$$x=\frac{\sqrt{6}}{2}+1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1-\frac{\sqrt{6}}{2} for x_{1} and 1+\frac{\sqrt{6}}{2} for x_{2}.

$$-2x^{2}+4x+1=-2\left(x-\left(-\frac{\sqrt{6}}{2}+1\right)\right)\left(x-\left(\frac{\sqrt{6}}{2}+1\right)\right)$$

Evaluate

$$1+4x-2x^{2}$$