Solve c(x)=2x^2-40x+2405

$$c ( x ) = 2 x ^ { 2 } - 40 x + 2405$$

Evaluate

$$2x^{2}-40x+2405$$

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

$$x ^ 2 -20x +\frac{2405}{2} = 0$$

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

$$r + s = 20 rs = \frac{2405}{2}$$

Two numbers r and s sum up to 20 exactly when the average of the two numbers is \frac{1}{2}*20 = 10. 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>

$$r = 10 - u s = 10 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{2405}{2}

$$(10 - u) (10 + u) = \frac{2405}{2}$$

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

$$100 - u^2 = \frac{2405}{2}$$

Simplify the expression by subtracting 100 on both sides

$$-u^2 = \frac{2405}{2}-100 = \frac{2205}{2}$$

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

$$u^2 = -\frac{2205}{2} u = \pm\sqrt{-\frac{2205}{2}} = \pm \frac{\sqrt{2205}}{\sqrt{2}}i$$

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

$$r =10 - \frac{\sqrt{2205}}{\sqrt{2}}i = 10 - 33.204i s = 10 + \frac{\sqrt{2205}}{\sqrt{2}}i = 10 + 33.204i$$

Differentiate w.r.t. x

$$4\left(x-10\right)$$

Steps Using Derivative Rule for Sum

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$2\times 2x^{2-1}-40x^{1-1}$$

Multiply 2 times 2.

$$4x^{2-1}-40x^{1-1}$$

Subtract 1 from 2.

$$4x^{1}-40x^{1-1}$$

Subtract 1 from 1.

$$4x^{1}-40x^{0}$$

For any term t, t^{1}=t.

$$4x-40x^{0}$$

For any term t except 0, t^{0}=1.

$$4x-40$$