Solve 3x^2+12x+36

$$3 x ^ { 2 } + 12 x + 36$$

Factor

$$3\left(x^{2}+4x+12\right)$$

Solution Steps

Factor out 3. Polynomial x^{2}+4x+12 is not factored since it does not have any rational roots.

$$3\left(x^{2}+4x+12\right)$$

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.

$$3x^{2}+12x+36=0$$

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.

$$x=\frac{-12±\sqrt{12^{2}-4\times 3\times 36}}{2\times 3}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 3\times 36}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-12±\sqrt{144-12\times 36}}{2\times 3}$$

Multiply -12 times 36.

$$x=\frac{-12±\sqrt{144-432}}{2\times 3}$$

Add 144 to -432.

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

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

$$3x^{2}+12x+36$$

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

$$x ^ 2 +4x +12 = 0$$

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

$$r + s = -4 rs = 12$$

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 = 12

$$(-2 - u) (-2 + u) = 12$$

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

$$4 - u^2 = 12$$

Simplify the expression by subtracting 4 on both sides

$$-u^2 = 12-4 = 8$$

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

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

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

$$r =-2 - \sqrt{8}i s = -2 + \sqrt{8}i$$

Evaluate

$$3\left(x^{2}+4x+12\right)$$