Question

How do you factor `y= x^2-3x-1` ?

Answer 1

It requires irrational coefficients, but...

Complete the square and use the difference of squares identity to find:

`y = x^2-3x-1`

`= (x-3/2-sqrt(13)/2)(x-3/2+sqrt(13)/2)`

Explanation:

Note that:

`(x-3/2)^2 = x^2-3x+9/4`

Also, the difference of squares identity can be written:

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

We use this with `a = (x-3/2)` and `b=sqrt(13)/2` below...

`y = x^2-3x-1`

`= x^2-3x+9/4-9/4-1`

`= (x-3/2)^2-13/4`

`= (x-3/2)^2-(sqrt(13)/2)^2`

`= ((x-3/2)-sqrt(13)/2)((x-3/2)+sqrt(13)/2)`

`= (x-3/2-sqrt(13)/2)(x-3/2+sqrt(13)/2)`