Question

How do you factor the trinomial `4x^2 - 8x = -3`?

Answer 1

`color(green)(4x^2-8x=-3) hArr color(red)((2x-3)(2x-1)=0)`

Explanation:

Re-writing `4x^2-8x=-3` into a form suitable for factoring:
`color(white)("XXX")4x^2-8x+3=0`

We are looking for factors `a and b` of `4`
which can be combined with factors `c and d` of `3`

such that `axxd + cxxb = -8`

`color(white)("XXX")rarr (ax+c)(bx+d)=4x^2-8x+3`

Assuming `a and b` are positive, `c and d` must be negative.

Our only factor sets are:
`(a,b) in {(4,1), (2,2), (1,4)}`
and
`(c,d) in {(-1,-3), (-3,1)}`
`color(white)("XXX")`(assuming integer factors)

With this few possibilities to check we find:
`color(white)("XXX")(2,2) and (-3,-1)` fairly quickly
which gives the factors
`color(white)("XXX")(2x-3)(2x-1)`