Question

How do you factor `49x^2 - 14x - 3`?

Answer 1

Since `49x^2-14x-3` is a quadratic of the form `ax^2+bx+c`
we could use the formula (which you really should memorize)
`(-b +- sqrt(b^2-4ac))/(2a)`

But first let's see if there is a simpler method under the assumption that we are dealing solely with integers (not necessarily true, but the work is easier if it is true).

We are hoping to find something of the form
`(px+q)(rx+s) = 49x^2-14x-3`

If all values are integer there are only 3 possibilities for `(p,r)`
namely `(1,49)`, `(49,1)` and `{7,7}`

The `(1,49)` and `(49,1)` pairs don't look too likely given the relatively small coefficient of `x` (we'll return to them later if necessary), so let's try `(p,r) = (7,7)`

We are hoping to find integer values `q` and `s` such that
`(7x+q)(7x+s) = 49x^2-14x-3`

For this to work
`7q+7s = 14` which implies `q+s = 2`
and
`qs= -3`

Again with limited possibilities for integer values of `q` and `s` where `qs = -3`

We can quickly deduce that the required factorization is
`(7x-3)(7x+1)`