Question 590924


Looking at the expression {{{2x^2-5x-7}}}, we can see that the first coefficient is {{{2}}}, the second coefficient is {{{-5}}}, and the last term is {{{-7}}}.



Now multiply the first coefficient {{{2}}} by the last term {{{-7}}} to get {{{(2)(-7)=-14}}}.



Now the question is: what two whole numbers multiply to {{{-14}}} (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 {{{-14}}} (the previous product).



Factors of {{{-14}}}:

1,2,7,14

-1,-2,-7,-14



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



These factors pair up and multiply to {{{-14}}}.

1*(-14) = -14
2*(-7) = -14
(-1)*(14) = -14
(-2)*(7) = -14


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>-14</font></td><td  align="center"><font color=black>1+(-14)=-13</font></td></tr><tr><td  align="center"><font color=red>2</font></td><td  align="center"><font color=red>-7</font></td><td  align="center"><font color=red>2+(-7)=-5</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>14</font></td><td  align="center"><font color=black>-1+14=13</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>7</font></td><td  align="center"><font color=black>-2+7=5</font></td></tr></table>



From the table, we can see that the two numbers {{{2}}} and {{{-7}}} add to {{{-5}}} (the middle coefficient).



So the two numbers {{{2}}} and {{{-7}}} both multiply to {{{-14}}} <font size=4><b>and</b></font> add to {{{-5}}}



Now replace the middle term {{{-5x}}} with {{{2x-7x}}}. Remember, {{{2}}} and {{{-7}}} add to {{{-5}}}. So this shows us that {{{2x-7x=-5x}}}.



{{{2x^2+highlight(2x-7x)-7}}} Replace the second term {{{-5x}}} with {{{2x-7x}}}.



{{{(2x^2+2x)+(-7x-7)}}} Group the terms into two pairs.



{{{2x(x+1)+(-7x-7)}}} Factor out the GCF {{{2x}}} from the first group.



{{{2x(x+1)-7(x+1)}}} Factor out {{{7}}} 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.



{{{(2x-7)(x+1)}}} Combine like terms. Or factor out the common term {{{x+1}}}



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Answer:



So {{{2x^2-5x-7}}} factors to {{{(2x-7)(x+1)}}}.



In other words, {{{2x^2-5x-7=(2x-7)(x+1)}}}.



Note: you can check the answer by expanding {{{(2x-7)(x+1)}}} to get {{{2x^2-5x-7}}} or by graphing the original expression and the answer (the two graphs should be identical).


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