Question

How do you factor `3x^2-22x-16`?

Answer 1

Factor by grouping. `(3x+2)(x-8)`

Explanation:

First multiply your `a and b` terms to get `-48`
Then find two numbers that multiply to that and add up to `-22`
Once you do that rewrite you `b` term as those two numbers like so :
`3x^2 +2x - 24x -16`
Then put parenthesis around your first and last two terms so you can factor them:
`(3x^2 +2x) (- 24x -16)`
Then factor normally:
`x(3x+2) -8(3x+2)`
your two parenthesis should be the same and if they are cross out one of them and make a new parenthesis with the two numbers you factored out:
`(3x+2)(x-8)`

Answer 2

(3x+2)(x-8)

Explanation:

We begin with `3x^2-22x-16` Automatically we can set up this..
`(3x+?)(x+?)`Since `3x^2` could only factor into `3x` and `x`
Now to fill in the ?'s
The question marks must multiply to equal 16 and add up to -22.
You can always use the guess and check method until you come across the correct answer or set up some equations..
`3x+y=22`
`xy=-16`
Then to solve this system of equations you can use subsitution...

Answer 3

`(x-8)(3x+2)`

Explanation:

These are not always easy to spot. Sometimes you really have to 'play' with them to find the correct combination.

First lets look at the constant of `-16`

As this is negative it means that it is the product of a negative and a positive value.

What are all the whole number factors of 16?

`1xx16; 2xx8; 4xx4`

3 is a prime number so we can only have `1xx3` giving:

`(x+-?)(3x+-?)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
I am deliberately 'going round the houses' to make a point!

`color(brown)("Try 1 and 16 for "-22x)`

`(x+-?)(3x+-?) -> (x+-1)(3x+-16)`

`+-16+-3!=-22: color(white)("dd") +-(3xx16)+-1=-22 color(red)(larr"Fail")`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(brown)("Try 4 and 4 for "-22x)`

`(x+-?)(3x+-?) -> (x+-4)(3x+-4)`

`+-(3xx4)+-4!=-22 color(red)(larr" Fail")`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(brown)("Try 2 and 8 for "-22x)`

`(x+-?)(3x+-?) -> (x+-2)(3x+-8)`

`(3xx2)-8!=-22 color(red)(larr" Fail")`

`(3xx(-8))+2 =-22 color(green)(larr" WORKS!")`

`(x-8)(3x+2)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Check")`

`+x(3x+2) = 3x^2+2x`
`-8(3x+2)=ul(color(white)(ccd.) -24x-16)`
`color(white)("dddddddddddd")3x^2-22x-16 larr" As required"`