Question

How do you factor `6x^2-5x-4`?

Answer 1

`6x^2-5x-4 =(3x-4)(2x+1)`

Explanation:

The rule to factorise any quadratic is to find two numbers such that

`"product" = x^2 " coefficient "xx" constant coefficient"`
`"sum" \ \ \ \ \ \ = x " coefficient"`

So for `6x^2-5x-4` we seek two numbers such that

`"product" = (6)*(-4) = -24`
`"sum" \ \ \ \ \ \ = -5`

So we look at the factors of `-24`. As the sum is negative and the product is negative then one of the factors must be negative, We can check every combination of the product factors:

`{: ("factor1", "factor2", "sum"), (1,-24,-23),(2,-12,-10), (3,-8,-5) ,(4,-6,-2),(-1,24,23),(-2,12,10),(-3,8,5),(-4,6,2) :}`

So the factors we seek are `color(blue)(3)` and `color(green)(-8)`

Therefore we can factorise the quadratic as follows:

`6x^2-5x-4= 6x^2 color(blue)(+3)x color(green)(-8)x -4`
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= 3x(2x+1) -4(2x+1)`
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= (3x-4)(2x+1)`

Similarly if we grouped the factors the other way around we get the same answer:

`6x^2-5x-4= 6x^2 color(green)(-8)x color(blue)(+3)x-4`
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= 2x(3x-4) + (3x-4)`
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= (2x+1)(3x-4)`

This approach works for all quadratics (assuming it does factorise) , The middle step in the last section can usually be skipped with practice.