Solve f(x)=2x^2+2x-1

$$f ( x ) = 2 x ^ { 2 } + 2 x - 1$$

Factor

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

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.

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

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.

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

Square 2.

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

Multiply -4 times 2.

$$x=\frac{-2±\sqrt{4-8\left(-1\right)}}{2\times 2}$$

Multiply -8 times -1.

$$x=\frac{-2±\sqrt{4+8}}{2\times 2}$$

Add 4 to 8.

$$x=\frac{-2±\sqrt{12}}{2\times 2}$$

Take the square root of 12.

$$x=\frac{-2±2\sqrt{3}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-2±2\sqrt{3}}{4}$$

Now solve the equation x=\frac{-2±2\sqrt{3}}{4} when ± is plus. Add -2 to 2\sqrt{3}.

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

Divide -2+2\sqrt{3} by 4.

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

Now solve the equation x=\frac{-2±2\sqrt{3}}{4} when ± is minus. Subtract 2\sqrt{3} from -2.

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

Divide -2-2\sqrt{3} by 4.

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

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

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

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 2

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

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

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

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}{2}

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

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

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

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

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

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

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

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

$$r =-\frac{1}{2} - \frac{\sqrt{3}}{2} = -1.366 s = -\frac{1}{2} + \frac{\sqrt{3}}{2} = 0.366$$

Evaluate

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