Question 168715


{{{24x^2-10x-4}}} Start with the given expression



{{{2(12x^2-5x-2)}}} Factor out the GCF {{{2}}}



Now let's focus on the inner expression {{{12x^2-5x-2}}}





------------------------------------------------------------





Looking at the expression {{{12x^2-5x-2}}}, we can see that the first coefficient is {{{12}}}, the second coefficient is {{{-5}}}, and the last term is {{{-2}}}.



Now multiply the first coefficient {{{12}}} by the last term {{{-2}}} to get {{{(12)(-2)=-24}}}.



Now the question is: what two whole numbers multiply to {{{-24}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-5}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-24}}} (the previous product).



Factors of {{{-24}}}:

1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-24}}}.

1*(-24)
2*(-12)
3*(-8)
4*(-6)
(-1)*(24)
(-2)*(12)
(-3)*(8)
(-4)*(6)


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-5}}}:



<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=black>1</font></td><td  align="center"><font color=black>-24</font></td><td  align="center"><font color=black>1+(-24)=-23</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-12</font></td><td  align="center"><font color=black>2+(-12)=-10</font></td></tr><tr><td  align="center"><font color=red>3</font></td><td  align="center"><font color=red>-8</font></td><td  align="center"><font color=red>3+(-8)=-5</font></td></tr><tr><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>4+(-6)=-2</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>24</font></td><td  align="center"><font color=black>-1+24=23</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>12</font></td><td  align="center"><font color=black>-2+12=10</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>8</font></td><td  align="center"><font color=black>-3+8=5</font></td></tr><tr><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>-4+6=2</font></td></tr></table>



From the table, we can see that the two numbers {{{3}}} and {{{-8}}} add to {{{-5}}} (the middle coefficient).



So the two numbers {{{3}}} and {{{-8}}} both multiply to {{{-24}}} <font size=4><b>and</b></font> add to {{{-5}}}



Now replace the middle term {{{-5x}}} with {{{3x-8x}}}. Remember, {{{3}}} and {{{-8}}} add to {{{-5}}}. So this shows us that {{{3x-8x=-5x}}}.



{{{12x^2+highlight(3x-8x)-2}}} Replace the second term {{{-5x}}} with {{{3x-8x}}}.



{{{(12x^2+3x)+(-8x-2)}}} Group the terms into two pairs.



{{{3x(4x+1)+(-8x-2)}}} Factor out the GCF {{{3x}}} from the first group.



{{{3x(4x+1)-2(4x+1)}}} Factor out {{{2}}} from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



{{{(3x-2)(4x+1)}}} Combine like terms. Or factor out the common term {{{4x+1}}}






------------------------------------------------------------





So our expression goes from {{{2(12x^2-5x-2)}}} and factors further to {{{2(3x-2)(4x+1)}}}



------------------

Answer:


So {{{24x^2-10x-4}}} completely factors to {{{2(3x-2)(4x+1)}}}