Question

Is it possible to factor `y= x^2 -x - 11`? If so, what are the factors?

Answer 1

Yes, with irrational coefficients:

`x^2-x-11 = (x-1/2-(3sqrt(5))/2)(x-1/2+(3sqrt(5))/2)`

Explanation:

Complete the square and use the difference of squares identity:

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

with `a=(x-1/2)` and `b = sqrt(45)/2` ...

`x^2-x-11`

`=x^2-x+1/4-45/4`

`=(x-1/2)^2-45/4`

`=(x-1/2)^2-(sqrt(45)/2)^2`

`= ((x-1/2)-sqrt(45)/2)((x-1/2)+sqrt(45)/2)`

`= (x-1/2-sqrt(45)/2)(x-1/2+sqrt(45)/2)`

Finally note that if `a, b >= 0` then `sqrt(ab) = sqrt(a)sqrt(b)`, so we can simplify:

`sqrt(45) = sqrt(9*5) = sqrt(9)*sqrt(5) = 3sqrt(5)`

So we can write:

`x^2-x-11 = (x-1/2-(3sqrt(5))/2)(x-1/2+(3sqrt(5))/2)`