Question 422089


{{{12d^2+14d-6}}} Start with the given expression.



{{{2(6d^2+7d-3)}}} Factor out the GCF {{{2}}}.



Now let's try to factor the inner expression {{{6d^2+7d-3}}}



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Looking at the expression {{{6d^2+7d-3}}}, we can see that the first coefficient is {{{6}}}, the second coefficient is {{{7}}}, and the last term is {{{-3}}}.



Now multiply the first coefficient {{{6}}} by the last term {{{-3}}} to get {{{(6)(-3)=-18}}}.



Now the question is: what two whole numbers multiply to {{{-18}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{7}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-18}}} (the previous product).



Factors of {{{-18}}}:

1,2,3,6,9,18

-1,-2,-3,-6,-9,-18



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



These factors pair up and multiply to {{{-18}}}.

1*(-18) = -18
2*(-9) = -18
3*(-6) = -18
(-1)*(18) = -18
(-2)*(9) = -18
(-3)*(6) = -18


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{7}}}:



<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>-18</font></td><td  align="center"><font color=black>1+(-18)=-17</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-9</font></td><td  align="center"><font color=black>2+(-9)=-7</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>3+(-6)=-3</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>18</font></td><td  align="center"><font color=black>-1+18=17</font></td></tr><tr><td  align="center"><font color=red>-2</font></td><td  align="center"><font color=red>9</font></td><td  align="center"><font color=red>-2+9=7</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>-3+6=3</font></td></tr></table>



From the table, we can see that the two numbers {{{-2}}} and {{{9}}} add to {{{7}}} (the middle coefficient).



So the two numbers {{{-2}}} and {{{9}}} both multiply to {{{-18}}} <font size=4><b>and</b></font> add to {{{7}}}



Now replace the middle term {{{7d}}} with {{{-2d+9d}}}. Remember, {{{-2}}} and {{{9}}} add to {{{7}}}. So this shows us that {{{-2d+9d=7d}}}.



{{{6d^2+highlight(-2d+9d)-3}}} Replace the second term {{{7d}}} with {{{-2d+9d}}}.



{{{(6d^2-2d)+(9d-3)}}} Group the terms into two pairs.



{{{2d(3d-1)+(9d-3)}}} Factor out the GCF {{{2d}}} from the first group.



{{{2d(3d-1)+3(3d-1)}}} Factor out {{{3}}} 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.



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



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So {{{2(6d^2+7d-3)}}} then factors further to {{{2(2d+3)(3d-1)}}}



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Answer:



So {{{12d^2+14d-6}}} completely factors to {{{2(2d+3)(3d-1)}}}.



In other words, {{{12d^2+14d-6=2(2d+3)(3d-1)}}}.



Note: you can check the answer by expanding {{{2(2d+3)(3d-1)}}} to get {{{12d^2+14d-6}}} or by graphing the original expression and the answer (the two graphs should be identical).



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