Solve 8x^2-32x+16

$$8 x ^ { 2 } - 32 x + 16$$

Factor

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

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.

$$8x^{2}-32x+16=0$$

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.

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

Square -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\times 8\times 16}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-32\right)±\sqrt{1024-32\times 16}}{2\times 8}$$

Multiply -32 times 16.

$$x=\frac{-\left(-32\right)±\sqrt{1024-512}}{2\times 8}$$

Add 1024 to -512.

$$x=\frac{-\left(-32\right)±\sqrt{512}}{2\times 8}$$

Take the square root of 512.

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

The opposite of -32 is 32.

$$x=\frac{32±16\sqrt{2}}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{32±16\sqrt{2}}{16}$$

Now solve the equation x=\frac{32±16\sqrt{2}}{16} when ± is plus. Add 32 to 16\sqrt{2}.

$$x=\frac{16\sqrt{2}+32}{16}$$

Divide 32+16\sqrt{2} by 16.

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

Now solve the equation x=\frac{32±16\sqrt{2}}{16} when ± is minus. Subtract 16\sqrt{2} from 32.

$$x=\frac{32-16\sqrt{2}}{16}$$

Divide 32-16\sqrt{2} by 16.

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

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

$$8x^{2}-32x+16=8\left(x-\left(\sqrt{2}+2\right)\right)\left(x-\left(2-\sqrt{2}\right)\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 8

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

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = 4 rs = 2$$

Two numbers r and s sum up to 4 exactly when the average of the two numbers is \frac{1}{2}*4 = 2. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = 2 - u s = 2 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 2

$$(2 - u) (2 + u) = 2$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$4 - u^2 = 2$$

Simplify the expression by subtracting 4 on both sides

$$-u^2 = 2-4 = -2$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = 2 u = \pm\sqrt{2} = \pm \sqrt{2}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =2 - \sqrt{2} = 0.586 s = 2 + \sqrt{2} = 3.414$$

Evaluate

$$8\left(x^{2}-4x+2\right)$$