Question

How do you factor `x^2 - 40x + 300`?

Answer 1

`x^2 -40x +300 = (x-10)(x-30)`

Explanation:

"Find factors of 300 which add to 40...."

This is not as daunting as it feels at first.

If the factors of a number are written in order, the sum and difference get less as you get closer to the middle factors.

For example: the factors of 36 are:

`1" "2" "3" "4" "6" "9" "12" "18" "36`
`color(white)(xxxxxxxxx..xx)uarr`
`color(white)(xxxxxxxxxxxx)sqrt36`

The outermost factors: `1 and 36`
have the greatest sum (37) and the greatest difference (35)

`6` is exactly in the middle - it is `sqrt36`

The innermost factors `6 and 6` have the smallest sum (12)
and the smallest difference (0)

`x^2 -40x +300`

40 is not very big compared to 300.

This means that the factor pair we want is not very far from the middle of the list of factors.

`sqrt300 = 17.3....`

`40 div 2 = 20`

The factors we need will be less than 20 on each side of 17.

We find `30 xx 10 = 300 and 30+10 = 40" "` BINGO!!

The signs will both be negative.

`x^2 -40x +300 = (x-10)(x-30)`