Question

How do you factor `1 - 4b² + a² - 2ab`?

Answer 1

This expression does not factorise.

Explanation:

Given: `1-4b^2+a^2-2ab`

Notice that the expression is a mixture of terms of degree `2` and degree `0`.

If it did factor then the factors would be a mixture of terms of degree `1` and `0`.

The factors can be scaled so that the coefficient of `a` is `1` in both factors, resulting in a factorisation of the form:

`1-4b^2+a^2-2ab`

`=(a+pb+r)(a+qb+s)`

`=a^2+(p+q)ab+pqb^2+(r+s)a+(ps+qr)b+rs`

Equating coefficients, we find:

`{(p+q=-2), (pq=-4), (r+s=0), (ps+qr=0), (rs=1) :}`

From the first two equations, we find that `p` and `q` are (in some order):

`{ (-1+sqrt(5)), (-1-sqrt(5)) :}`

From the third and fifth equation, we find that `r` and `s` are (in some order):

`{ (i), (-i) :}`

Then we find that:

`ps+qr = +-2sqrt(5)i`

contradicting the fourth equation.

So there is no such factorisation.