Solve 4x^2-4x-1

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

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.

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

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 4\left(-1\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-4\right)±\sqrt{16-16\left(-1\right)}}{2\times 4}$$

Multiply -16 times -1.

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

Add 16 to 16.

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

Take the square root of 32.

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

The opposite of -4 is 4.

$$x=\frac{4±4\sqrt{2}}{2\times 4}$$

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.

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

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

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

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

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

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}.

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

Evaluate

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