Question

How do you factor completely: `3x^2 + 12x + 7`?

Answer 1

It depends on what kinds of numbers we are using.

Explanation:

`3x^2 + 12x + 7`

Using integers (or rational numbers) this quadratic cannot be factored any further. It is already factored completely.
(The only possibilities are `(3x+1)(x+7)` and `(3x+7)(x+1)`, neither of which works.)

If we are using Real numbers or Complex numbers, we can factor the quadratic by finding its zeros.

Solve
`3x^2 + 12x + 7 = 0` using your choice of completing the square of the quadratic formula.

`x = (-(12) +- sqrt ((12)^2-4(3)(7)))/(2(3))`

`= (-12 +- sqrt (144-84))/6 = (-12 +- sqrt 60)/6 = (-12 +- 2sqrt 15)/6`

`= -2 +- sqrt 15/3`

The zeros are:

`z_1 = -2 + sqrt 15/3` and z_2 = -2 - sqrt 15/3#

The factors are:

`(x-z_1)(x-z_2) = (x -(-2 + sqrt 15/3))(x- (-2 - sqrt 15/3))`

`= (x + 2 - sqrt 15/3))(x + 2 + sqrt 15/3))`