Question

Is it possible to factor `y=x^2-4x-3`? If so, what are the factors?

Answer 1

Factors are `y=(x-2+sqrt7)(x-2-sqrt7)`

Explanation:

When you generally say - Ïs it possible to factor `y=ax^2+bx+c`, what is normally meant is to have rational factors of type `y=a(x+p)(x+q)`

This is possible if the discriminant `b^2-4ac` is square of a rational number.

Here in `y=x^2-4x-3`, we have `a=1`, `b=-4` and `c=-3` and hence `b^2-4ac=(-4)^2-4xx1xx(-3)=16+12=28`

as it is not the square of a rational number you cannot have rational factors. But it is still possible to have irrational factors.

As `y=x^2-4x-3=(x^2-2xx2xx x+2^2)-2^2-3`

= `(x-2)^2-7=(x-2)^2-(sqrt7)^2`

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

factors are `y=(x-2+sqrt7)(x-2-sqrt7)`.