Solve 3x^2+9x+12

$$3 { x }^{ 2 } +9x+12$$

Factor

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

Solution Steps

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

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

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.

$$3x^{2}+9x+12=0$$

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.

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

Square 9.

$$x=\frac{-9±\sqrt{81-4\times 3\times 12}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-9±\sqrt{81-12\times 12}}{2\times 3}$$

Multiply -12 times 12.

$$x=\frac{-9±\sqrt{81-144}}{2\times 3}$$

Add 81 to -144.

$$x=\frac{-9±\sqrt{-63}}{2\times 3}$$

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

$$3x^{2}+9x+12$$

Evaluate

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