Question 143268
Do you want to factor?





{{{6x^2-28x+16}}} Start with the given expression



{{{2(3x^2-14x+8)}}} Factor out the GCF {{{2}}}



Now let's focus on the inner expression {{{3x^2-14x+8}}}





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Looking at {{{3x^2-14x+8}}} we can see that the first term is {{{3x^2}}} and the last term is {{{8}}} where the coefficients are 3 and 8 respectively.


Now multiply the first coefficient 3 and the last coefficient 8 to get 24. Now what two numbers multiply to 24 and add to the  middle coefficient -14? Let's list all of the factors of 24:




Factors of 24:

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


-1,-2,-3,-4,-6,-8,-12,-24 ...List the negative factors as well. 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)


note: remember two negative numbers multiplied together make a positive number



Now which of these pairs add to -14? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -14


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">24</td><td>1+24=25</td></tr><tr><td align="center">2</td><td align="center">12</td><td>2+12=14</td></tr><tr><td align="center">3</td><td align="center">8</td><td>3+8=11</td></tr><tr><td align="center">4</td><td align="center">6</td><td>4+6=10</td></tr><tr><td align="center">-1</td><td align="center">-24</td><td>-1+(-24)=-25</td></tr><tr><td align="center">-2</td><td align="center">-12</td><td>-2+(-12)=-14</td></tr><tr><td align="center">-3</td><td align="center">-8</td><td>-3+(-8)=-11</td></tr><tr><td align="center">-4</td><td align="center">-6</td><td>-4+(-6)=-10</td></tr></table>



From this list we can see that -2 and -12 add up to -14 and multiply to 24



Now looking at the expression {{{3x^2-14x+8}}}, replace {{{-14x}}} with {{{-2x+-12x}}} (notice {{{-2x+-12x}}} adds up to {{{-14x}}}. So it is equivalent to {{{-14x}}})


{{{3x^2+highlight(-2x+-12x)+8}}}



Now let's factor {{{3x^2-2x-12x+8}}} by grouping:



{{{(3x^2-2x)+(-12x+8)}}} Group like terms



{{{x(3x-2)-4(3x-2)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-4}}} out of the second group



{{{(x-4)(3x-2)}}} Since we have a common term of {{{3x-2}}}, we can combine like terms


So {{{3x^2-2x-12x+8}}} factors to {{{(x-4)(3x-2)}}}



So this also means that {{{3x^2-14x+8}}} factors to {{{(x-4)(3x-2)}}} (since {{{3x^2-14x+8}}} is equivalent to {{{3x^2-2x-12x+8}}})




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So our expression goes from {{{2(3x^2-14x+8)}}} and factors further to {{{2(x-4)(3x-2)}}}



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Answer:


So {{{6x^2-28x+16}}} factors to {{{2(x-4)(3x-2)}}}