Solve 8x^2-16x+4

$$8 { x }^{ 2 } -16x+4$$

Factor

$$8\left(x-\left(-\frac{\sqrt{2}}{2}+1\right)\right)\left(x-\left(\frac{\sqrt{2}}{2}+1\right)\right)$$

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.

$$8x^{2}-16x+4=0$$

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.

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

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 8\times 4}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-16\right)±\sqrt{256-32\times 4}}{2\times 8}$$

Multiply -32 times 4.

$$x=\frac{-\left(-16\right)±\sqrt{256-128}}{2\times 8}$$

Add 256 to -128.

$$x=\frac{-\left(-16\right)±\sqrt{128}}{2\times 8}$$

Take the square root of 128.

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

The opposite of -16 is 16.

$$x=\frac{16±8\sqrt{2}}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{16±8\sqrt{2}}{16}$$

Now solve the equation x=\frac{16±8\sqrt{2}}{16} when ± is plus. Add 16 to 8\sqrt{2}.

$$x=\frac{8\sqrt{2}+16}{16}$$

Divide 16+8\sqrt{2} by 16.

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

Now solve the equation x=\frac{16±8\sqrt{2}}{16} when ± is minus. Subtract 8\sqrt{2} from 16.

$$x=\frac{16-8\sqrt{2}}{16}$$

Divide 16-8\sqrt{2} by 16.

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

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

$$8x^{2}-16x+4=8\left(x-\left(\frac{\sqrt{2}}{2}+1\right)\right)\left(x-\left(-\frac{\sqrt{2}}{2}+1\right)\right)$$

Evaluate

$$8x^{2}-16x+4$$