Question 607717


Looking at the expression {{{12d^2+4d-1}}}, we can see that the first coefficient is {{{12}}}, the second coefficient is {{{4}}}, and the last term is {{{-1}}}.



Now multiply the first coefficient {{{12}}} by the last term {{{-1}}} to get {{{(12)(-1)=-12}}}.



Now the question is: what two whole numbers multiply to {{{-12}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{4}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-12}}} (the previous product).



Factors of {{{-12}}}:

1,2,3,4,6,12

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



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



These factors pair up and multiply to {{{-12}}}.

1*(-12) = -12
2*(-6) = -12
3*(-4) = -12
(-1)*(12) = -12
(-2)*(6) = -12
(-3)*(4) = -12


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{4}}}:



<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>-12</font></td><td  align="center"><font color=black>1+(-12)=-11</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>2+(-6)=-4</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>3+(-4)=-1</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>12</font></td><td  align="center"><font color=black>-1+12=11</font></td></tr><tr><td  align="center"><font color=red>-2</font></td><td  align="center"><font color=red>6</font></td><td  align="center"><font color=red>-2+6=4</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>-3+4=1</font></td></tr></table>



From the table, we can see that the two numbers {{{-2}}} and {{{6}}} add to {{{4}}} (the middle coefficient).



So the two numbers {{{-2}}} and {{{6}}} both multiply to {{{-12}}} <font size=4><b>and</b></font> add to {{{4}}}



Now replace the middle term {{{4d}}} with {{{-2d+6d}}}. Remember, {{{-2}}} and {{{6}}} add to {{{4}}}. So this shows us that {{{-2d+6d=4d}}}.



{{{12d^2+highlight(-2d+6d)-1}}} Replace the second term {{{4d}}} with {{{-2d+6d}}}.



{{{(12d^2-2d)+(6d-1)}}} Group the terms into two pairs.



{{{2d(6d-1)+(6d-1)}}} Factor out the GCF {{{2d}}} from the first group.



{{{2d(6d-1)+1(6d-1)}}} Factor out {{{1}}} 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+1)(6d-1)}}} Combine like terms. Or factor out the common term {{{6d-1}}}



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Answer:



So {{{12d^2+4d-1}}} factors to {{{(2d+1)(6d-1)}}}.



In other words, {{{12d^2+4d-1=(2d+1)(6d-1)}}}.



Note: you can check the answer by expanding {{{(2d+1)(6d-1)}}} to get {{{12d^2+4d-1}}} or by graphing the original expression and the answer (the two graphs should be identical).


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