Question

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

Answer 1

mathematics is the study of patterns and you have two in this problem.

Explanation:

One of the accepted definitions of mathematics is that it is the study of patterns. We need to be able to recognize these patterns to simplify our lives.

The first pattern is the greatest common factor. In this problem we should see that 3 is common to both 3 and 12. It should then be factored out immediately.

`3(x^2-4)`

this leaves us with `(x^2-4)` inside the parenthesis. This too is a pattern. It is the difference of two squares. We can take the square root of both:
`sqrt(x^2) = x` and `sqrt(4) = 2`

so the pattern for the difference of two squares is:

`a^2-b^2 = (a+b)(a-b)`

then `(x^2-4)` becomes:

`(x+2)(x-2)`

bringing down the 3, we obtain our final solution:

`3(x+2)(x-2)`

which is completely factored and that makes us happy.