Solve {l}{3x^2+12}{+6x}

$$\left. \begin{array} { l } { 3 x ^ { 2 } + 12 } \\ { + 6 x } \end{array} \right.$$

Factor

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

Solution Steps

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

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

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.

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

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.

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

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 3\times 12}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-6±\sqrt{36-12\times 12}}{2\times 3}$$

Multiply -12 times 12.

$$x=\frac{-6±\sqrt{36-144}}{2\times 3}$$

Add 36 to -144.

$$x=\frac{-6±\sqrt{-108}}{2\times 3}$$

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

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

Evaluate

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