Solve 12x^2+3x+6

$$12 x ^ { 2 } + 3 x + 6$$

Factor

$$3\left(4x^{2}+x+2\right)$$

Solution Steps

Factor out 3. Polynomial 4x^{2}+x+2 is not factored since it does not have any rational roots.

$$3\left(4x^{2}+x+2\right)$$

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.

$$12x^{2}+3x+6=0$$

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.

$$x=\frac{-3±\sqrt{3^{2}-4\times 12\times 6}}{2\times 12}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\times 12\times 6}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-3±\sqrt{9-48\times 6}}{2\times 12}$$

Multiply -48 times 6.

$$x=\frac{-3±\sqrt{9-288}}{2\times 12}$$

Add 9 to -288.

$$x=\frac{-3±\sqrt{-279}}{2\times 12}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Quadratic polynomial cannot be factored.

$$12x^{2}+3x+6$$

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 12

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

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

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

Two numbers r and s sum up to -\frac{1}{4} exactly when the average of the two numbers is \frac{1}{2}*-\frac{1}{4} = -\frac{1}{8}. 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>

$$r = -\frac{1}{8} - u s = -\frac{1}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{1}{2}

$$(-\frac{1}{8} - u) (-\frac{1}{8} + u) = \frac{1}{2}$$

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

$$\frac{1}{64} - u^2 = \frac{1}{2}$$

Simplify the expression by subtracting \frac{1}{64} on both sides

$$-u^2 = \frac{1}{2}-\frac{1}{64} = \frac{31}{64}$$

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

$$u^2 = -\frac{31}{64} u = \pm\sqrt{-\frac{31}{64}} = \pm \frac{\sqrt{31}}{8}i$$

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

$$r =-\frac{1}{8} - \frac{\sqrt{31}}{8}i = -0.125 - 0.696i s = -\frac{1}{8} + \frac{\sqrt{31}}{8}i = -0.125 + 0.696i$$

Evaluate

$$3\left(4x^{2}+x+2\right)$$