Solve x^2-2x-1

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

Factor

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

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.

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

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.

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

Square -2.

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

Multiply -4 times -1.

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

Add 4 to 4.

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

Take the square root of 8.

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

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{2}}{2}$$

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

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

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

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

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

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

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

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

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

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

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.

$$x ^ 2 -2x -1 = 0$$

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

$$r + s = 2 rs = -1$$

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

$$r = 1 - u s = 1 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -1

$$(1 - u) (1 + u) = -1$$

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

$$1 - u^2 = -1$$

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -1-1 = -2$$

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.

$$r =1 - \sqrt{2} = -0.414 s = 1 + \sqrt{2} = 2.414$$

Evaluate

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