Solve 16x^2-16x+1

$$16 x ^ { 2 } - 16 x + 1$$

Factor

$$16\left(x-\left(-\frac{\sqrt{3}}{4}+\frac{1}{2}\right)\right)\left(x-\left(\frac{\sqrt{3}}{4}+\frac{1}{2}\right)\right)$$

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.

$$16x^{2}-16x+1=0$$

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.

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

Square -16.

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

Multiply -4 times 16.

$$x=\frac{-\left(-16\right)±\sqrt{256-64}}{2\times 16}$$

Add 256 to -64.

$$x=\frac{-\left(-16\right)±\sqrt{192}}{2\times 16}$$

Take the square root of 192.

$$x=\frac{-\left(-16\right)±8\sqrt{3}}{2\times 16}$$

The opposite of -16 is 16.

$$x=\frac{16±8\sqrt{3}}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{16±8\sqrt{3}}{32}$$

Now solve the equation x=\frac{16±8\sqrt{3}}{32} when ± is plus. Add 16 to 8\sqrt{3}.

$$x=\frac{8\sqrt{3}+16}{32}$$

Divide 16+8\sqrt{3} by 32.

$$x=\frac{\sqrt{3}}{4}+\frac{1}{2}$$

Now solve the equation x=\frac{16±8\sqrt{3}}{32} when ± is minus. Subtract 8\sqrt{3} from 16.

$$x=\frac{16-8\sqrt{3}}{32}$$

Divide 16-8\sqrt{3} by 32.

$$x=-\frac{\sqrt{3}}{4}+\frac{1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{2}+\frac{\sqrt{3}}{4} for x_{1} and \frac{1}{2}-\frac{\sqrt{3}}{4} for x_{2}.

$$16x^{2}-16x+1=16\left(x-\left(\frac{\sqrt{3}}{4}+\frac{1}{2}\right)\right)\left(x-\left(-\frac{\sqrt{3}}{4}+\frac{1}{2}\right)\right)$$

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 16

$$x ^ 2 -1x +\frac{1}{16} = 0$$

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

$$r + s = 1 rs = \frac{1}{16}$$

Two numbers r and s sum up to 1 exactly when the average of the two numbers is \frac{1}{2}*1 = \frac{1}{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>

$$r = \frac{1}{2} - u s = \frac{1}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{1}{16}

$$(\frac{1}{2} - u) (\frac{1}{2} + u) = \frac{1}{16}$$

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

$$\frac{1}{4} - u^2 = \frac{1}{16}$$

Simplify the expression by subtracting \frac{1}{4} on both sides

$$-u^2 = \frac{1}{16}-\frac{1}{4} = -\frac{3}{16}$$

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

$$u^2 = \frac{3}{16} u = \pm\sqrt{\frac{3}{16}} = \pm \frac{\sqrt{3}}{4}$$

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

$$r =\frac{1}{2} - \frac{\sqrt{3}}{4} = 0.067 s = \frac{1}{2} + \frac{\sqrt{3}}{4} = 0.933$$

Evaluate

$$16x^{2}-16x+1$$