Question

How do you factor `15b^2 - 7b - 2`?

Answer 1

`(3b-2)(5b+1)`

Explanation:

We have a quadratic expression in the form

`ax^2+bx+c`

`a=15`
`b=-7`
`c=-2`

To factor this, first find two numbers that multiply to give `ac` and add to give `b`.

For `ac=-30` and `b=-7`

Maybe you can see by inspection that these two numbers are:

`-10` and `3`

How did I see that? Look at the factors of `30`

`1,2,3,5,6,10,15,30`

Now try adding or subtracting them to get `-7`

Remember that one of them (and ONLY one of them) must be negative so that we get `-30` when we multiply them.

Now it's much easier to see that `-10` and `3` are the numbers we want.

So why did we do this? Our original expression was

`15b^2-7b-2`

Let's now split the middle term using the numbers we just found.

`rArr15b^2` `color(blue)(-10b)` `color(red)(+3b)-2`

Now factor `5b` from the first two terms.

`5b(3b-2)+(3b-2)`

Notice now that we have a common factor of `(3b-2)`.
Let's factor it out:

`(3b-2)(5b+1)`

And we're done!

NOTE:

It still would have worked if we had chosen to split the terms in the opposite order. Let's check:

`rArr15b^2` `color(red)(+3b)` `color(blue)(-10b)-2`

`rArr3b(5b+1)-2(5b+1)`

`rArr(5b+1)(3b-2)`