Solve -4x^2+1+4x

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

Factor

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

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.

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

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.

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

Square 4.

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

Multiply -4 times -4.

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

Add 16 to 16.

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

Take the square root of 32.

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

Multiply 2 times -4.

$$x=\frac{-4±4\sqrt{2}}{-8}$$

Now solve the equation x=\frac{-4±4\sqrt{2}}{-8} when ± is plus. Add -4 to 4\sqrt{2}.

$$x=\frac{4\sqrt{2}-4}{-8}$$

Divide -4+4\sqrt{2} by -8.

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

Now solve the equation x=\frac{-4±4\sqrt{2}}{-8} when ± is minus. Subtract 4\sqrt{2} from -4.

$$x=\frac{-4\sqrt{2}-4}{-8}$$

Divide -4-4\sqrt{2} by -8.

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

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

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

Evaluate

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