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

3 Answers
Apr 17, 2018

Answer:

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)#

Apr 17, 2018

Answer:

(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...

Apr 17, 2018

Answer:

#(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"#