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

Apr 17, 2018

Factor by grouping. $\left(3 x + 2\right) \left(x - 8\right)$

#### Explanation:

First multiply your $a \mathmr{and} b$ 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 :
$3 {x}^{2} + 2 x - 24 x - 16$
Then put parenthesis around your first and last two terms so you can factor them:
$\left(3 {x}^{2} + 2 x\right) \left(- 24 x - 16\right)$
Then factor normally:
$x \left(3 x + 2\right) - 8 \left(3 x + 2\right)$
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:
$\left(3 x + 2\right) \left(x - 8\right)$

Apr 17, 2018

(3x+2)(x-8)

#### Explanation:

We begin with $3 {x}^{2} - 22 x - 16$ Automatically we can set up this..
(3x+?)(x+?) Since $3 {x}^{2}$ could only factor into $3 x$ 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..
$3 x + y = 22$
$x y = - 16$
Then to solve this system of equations you can use subsitution...

Apr 17, 2018

$\left(x - 8\right) \left(3 x + 2\right)$

#### 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 $1 \times 3$ giving:

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

$\textcolor{b r o w n}{\text{Try 1 and 16 for } - 22 x}$

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

$\pm 16 \pm 3 \ne - 22 : \textcolor{w h i t e}{\text{dd") +-(3xx16)+-1=-22 color(red)(larr"Fail}}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b r o w n}{\text{Try 4 and 4 for } - 22 x}$

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

$\pm \left(3 \times 4\right) \pm 4 \ne - 22 \textcolor{red}{\leftarrow \text{ Fail}}$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b r o w n}{\text{Try 2 and 8 for } - 22 x}$

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

$\left(3 \times 2\right) - 8 \ne - 22 \textcolor{red}{\leftarrow \text{ Fail}}$

$\left(3 \times \left(- 8\right)\right) + 2 = - 22 \textcolor{g r e e n}{\leftarrow \text{ WORKS!}}$

$\left(x - 8\right) \left(3 x + 2\right)$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Check}}$

$+ x \left(3 x + 2\right) = 3 {x}^{2} + 2 x$
$- 8 \left(3 x + 2\right) = \underline{\textcolor{w h i t e}{\mathcal{d} .} - 24 x - 16}$
color(white)("dddddddddddd")3x^2-22x-16 larr" As required"