Question

How do you factor `6x^2+x-1` ?

Answer 1

`6x^2+x-1 = (2x+1)(3x-1)`

Explanation:

Here are a couple of methods (in no particular order):

Method 1

Note that:

`(ax+1)(bx-1) = abx^2+(b-a)x-1`

Comparing with:

`6x^2+x-1`

we want to find `a, b` such that `ab=6` and `b-a = 1`

The values `a=2`, `b=3` work, so we find:

`6x^2+x-1 = (2x+1)(3x-1)`

`color(white)()`
Method 2 - Completing the square

To avoid much arithmetic with fractions, multiply first by `24 = 6*2^2` then divide by it at the end:

`24(6x^2+x-1) = 144x^2+24x-24`

`color(white)(24(6x^2+x-1)) = (12x)^2+2(12x)+1-25`

`color(white)(24(6x^2+x-1)) = (12x+1)^2-5^2`

`color(white)(24(6x^2+x-1)) = ((12x+1)-5)((12x+1)+5)`

`color(white)(24(6x^2+x-1)) = (12x-4)(12x+6)`

`color(white)(24(6x^2+x-1)) = (4(3x-1))(6(2x+1))`

`color(white)(24(6x^2+x-1)) = 24(3x-1)(2x+1)`

Divide both ends by `24` to get:

`6x^2+x-1 = (3x-1)(2x+1)`