Solve {c}{3x^2+2x}{+4}

$$\left. \begin{array} { c } { 3 x ^ { 2 } + 2 x } \\ { + 4 } \end{array} \right.$$

Evaluate

$$3x^{2}+2x+4$$

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 3

$$x ^ 2 +\frac{2}{3}x +\frac{4}{3} = 0$$

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

$$r + s = -\frac{2}{3} rs = \frac{4}{3}$$

Two numbers r and s sum up to -\frac{2}{3} exactly when the average of the two numbers is \frac{1}{2}*-\frac{2}{3} = -\frac{1}{3}. 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>

$$r = -\frac{1}{3} - u s = -\frac{1}{3} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{4}{3}

$$(-\frac{1}{3} - u) (-\frac{1}{3} + u) = \frac{4}{3}$$

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

$$\frac{1}{9} - u^2 = \frac{4}{3}$$

Simplify the expression by subtracting \frac{1}{9} on both sides

$$-u^2 = \frac{4}{3}-\frac{1}{9} = \frac{11}{9}$$

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

$$u^2 = -\frac{11}{9} u = \pm\sqrt{-\frac{11}{9}} = \pm \frac{\sqrt{11}}{3}i$$

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

$$r =-\frac{1}{3} - \frac{\sqrt{11}}{3}i = -0.333 - 1.106i s = -\frac{1}{3} + \frac{\sqrt{11}}{3}i = -0.333 + 1.106i$$

Differentiate w.r.t. x

$$6x+2$$

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}.

$$2\times 3x^{2-1}+2x^{1-1}$$

Multiply 2 times 3.

$$6x^{2-1}+2x^{1-1}$$

Subtract 1 from 2.

$$6x^{1}+2x^{1-1}$$

Subtract 1 from 1.

$$6x^{1}+2x^{0}$$

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

$$6x+2x^{0}$$

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

$$6x+2\times 1$$

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

$$6x+2$$