Question

How do you factor completely: `y² -  12y  +  32`?

Answer 1

`(y-4)(y-8)`

Explanation:

because the middle sign is a minus
and the last sign is an addition, both the signs in the parenthesis would be minuses
`(y-?)(y-?)`

now the two "`?`" numbers
will be a pair of factors of `32`
and add up to `12`

so let's list the pairs of factors of `32` and what they add up to :)
`1 and 32 -> 33` [X]
`2 and 16 -> 18` [X]
`4 and 8 -> 12` [`sqrt`]

so it looks like the factor pair of `4 and 8` works!

we would just substitute the two numbers in for the two "`?`"
and get
`(y-4)(y-8)`

Answer 2

Solve f(y) = y^2 - 12y + 32

Ans: (y - 4)(y - 8)

Explanation:

Find 2 numbers p and q knowing sum (-12) and product (32).
p and q have same sign, since a and c same sign.
Factor pairs of 32 --> (2, 16)(4, 8). This sum is 12 = - b.
Then, p = -4 and q = -8 (their sum must be -12, and not 12)

f(y) = (y - 4)(y - 8)