Question

How do you factor `y= 8m^2 - 41m - 42`?

Answer 1

Find a suitable splitting of the middle term, then factor by grouping to find:

`8m^2-41m-42=(8m+7)(m-6)`

Explanation:

Find a pair of factors of `AC = 8*42 = 16*21 = 336` which differ by `B=41`

The split of `336` into a pair of factors must put all of the powers of `2` on one side, since the difference (`41`) is odd. If both factors were even, then the difference would be even too.

That leads to the following possibilities to consider:

`16xx21`

`bb (48xx7)`

`112xx3`

`336xx1`

Having found the pair `48`, `7` use that to split the middle term and factor by grouping:

`8m^2-41m-42`

`=8m^2-48m+7m-42`

`=(8m^2-48m)+(7m-42)`

`=8m(m-6)+7(m-6)`

`=(8m+7)(m-6)`

Answer 2

`y=(8x+7)(x-6)`

Explanation:

You could look for values `p, q, r, s` such that
`color(white)("XXX")pxxr=8`
`color(white)("XXX")qxxs=-42`
`color(white)("XXX")ps+qr=-41`
(perhaps using the AC method)

...but unless you get lucky, there are quite a few factorings possible.

As an alternative you could use the quadratic formula:
`color(white)("XXX")(-b+-sqrt(b^2-4ac))/(2a)`

The numbers involved are still ugly but if you use a calculator or spreadsheet (evaluating only the `+` of the `+-`)
you should get:
`color(white)("XXX")x=6` as a zero for this expression.

Therefore one of the factors will be:
`color(white)("XXX")(x-6)`
Simple division (`8x^2divx=8`) and (`(-42)div(-6))=+7`)
gives the other term:
`color(white)("XXX")(8x+7)`

Answer 3

Alternatively, complete the square to find:

`8m^2-41m-42 = (m-6)(8m+7)`

Explanation:

Alternatively, you can complete the square to proceed directly to the answer as follows:

`8m^2-41m-42`

`=8(m^2-41/8 m - 21/4)`

`=8(m^2-41/8 m + (41/16)^2 - (41/16)^2 - 21/4)`

`=8((m-41/16)^2 - 1681/256 - 1344/256)`

`=8((m-41/16)^2 - 3025/256)`

`=8((m-41/16)^2 - (55/16)^2)`

`=8((m-41/16) - 55/16)((m-41/16) + 55/16)`

`=8(m-96/16)(m+14/16)`

`=8(m-6)(m+7/8)`

`=(m-6)(8m+7)`

...using the difference of squares identity:

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

with `a = m-41/16` and `b = 55/16`

Ouch!