Solve f(x)=-4x^2+16x+2

$$f ( x ) = - 4 x ^ { 2 } + 16 x + 2$$

Factor

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

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.

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

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.

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

Square 16.

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

Multiply -4 times -4.

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

Multiply 16 times 2.

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

Add 256 to 32.

$$x=\frac{-16±\sqrt{288}}{2\left(-4\right)}$$

Take the square root of 288.

$$x=\frac{-16±12\sqrt{2}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-16±12\sqrt{2}}{-8}$$

Now solve the equation x=\frac{-16±12\sqrt{2}}{-8} when ± is plus. Add -16 to 12\sqrt{2}.

$$x=\frac{12\sqrt{2}-16}{-8}$$

Divide -16+12\sqrt{2} by -8.

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

Now solve the equation x=\frac{-16±12\sqrt{2}}{-8} when ± is minus. Subtract 12\sqrt{2} from -16.

$$x=\frac{-12\sqrt{2}-16}{-8}$$

Divide -16-12\sqrt{2} by -8.

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

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

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

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.

$$x ^ 2 -4x -\frac{1}{2} = 0$$

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

$$r + s = 4 rs = -\frac{1}{2}$$

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 = -\frac{1}{2}

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

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

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

Simplify the expression by subtracting 4 on both sides

$$-u^2 = -\frac{1}{2}-4 = -\frac{9}{2}$$

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

$$u^2 = \frac{9}{2} u = \pm\sqrt{\frac{9}{2}} = \pm \frac{3}{\sqrt{2}}$$

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

$$r =2 - \frac{3}{\sqrt{2}} = -0.121 s = 2 + \frac{3}{\sqrt{2}} = 4.121$$

Evaluate

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