Solve 256x^2+19x+15=

$$256 x ^ { 2 } + 19 x + 15 =$$

Evaluate

$$256x^{2}+19x+15$$

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 256

$$x ^ 2 +\frac{19}{256}x +\frac{15}{256} = 0$$

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

$$r + s = -\frac{19}{256} rs = \frac{15}{256}$$

Two numbers r and s sum up to -\frac{19}{256} exactly when the average of the two numbers is \frac{1}{2}*-\frac{19}{256} = -\frac{19}{512}. 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>

$$r = -\frac{19}{512} - u s = -\frac{19}{512} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{15}{256}

$$(-\frac{19}{512} - u) (-\frac{19}{512} + u) = \frac{15}{256}$$

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

$$\frac{361}{262144} - u^2 = \frac{15}{256}$$

Simplify the expression by subtracting \frac{361}{262144} on both sides

$$-u^2 = \frac{15}{256}-\frac{361}{262144} = -\frac{14999}{262144}$$

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

$$u^2 = \frac{14999}{262144} u = \pm\sqrt{\frac{14999}{262144}} = \pm \frac{\sqrt{14999}}{512}$$

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

$$r =-\frac{19}{512} - \frac{\sqrt{14999}}{512} = -0.037 - 0.239i s = -\frac{19}{512} + \frac{\sqrt{14999}}{512} = -0.037 + 0.239i$$

Differentiate w.r.t. x

$$512x+19$$

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}.

$$2\times 256x^{2-1}+19x^{1-1}$$

Multiply 2 times 256.

$$512x^{2-1}+19x^{1-1}$$

Subtract 1 from 2.

$$512x^{1}+19x^{1-1}$$

Subtract 1 from 1.

$$512x^{1}+19x^{0}$$

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

$$512x+19x^{0}$$

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

$$512x+19\times 1$$

For any term t, t\times 1=t and 1t=t.

$$512x+19$$