Question

How do you factor `5x^2-22x+8`?

Answer 1

Factor: y = 5x^2 - 22x + 8

Ans: (5x - 2)(x - 4)

Explanation:

Use the new AC Method to factor trinomial (Socratic Search):
`y = 5x^2 - 22x + 8 =` 5(x + p)(x + q)
Converted trinomial:
`y' = x^2 - 22x + 40 =` (x + p')(x + q')
Find p' and q', that have same sign, knowing sum (b = -22) and product (ac = 40). They are: p' = - 2 and q' = - 20
Therefor: `p = (p')/a = - 2/5` and `q = (q')/a = - 20/5 = - 4`
Factored form of y:
`y = 5(x - 2/5)(x - 4) = (5x - 2)(x - 4)`

Answer 2

`(x-4)(5x-2)`

Explanation:

`5x^2 color(blue)(-22)xcolor(red)(+)8`

Find factors of `5 and 8` whose products `color(red)("ADD")` to `color(blue)(22)`

`5` is a prime number, so the factors are only `1 and 5`

The factors of `8` are `1 xx 8` or the other pair `2 xx 4`

It may take a bit of trial and error, but notice that `5 xx 4 =20` which is close to the `20` we want.

`" "5 and 8" "` write the factors and cross multiply
`" "darr" "darr`
`" "1color(white)(xxxx)4" "rarr 5 xx 4 =20`
`" "5color(white)(xxxx)2" "rarr 1 xx 2 = ul2`
`color(white)(xxxxxxxxxxxxxxxxxxx)color(blue)(22)" "larr` the combination is correct.

The positive sign of `color(red)(+)8` indicates the following:

ADD the factors and the signs will be the SAME

The negative sign of `color(blue)(-22)` shows what the signs are.

The factors are `(1x-4)(5x-2)`