Question 560571


Looking at the expression {{{x^2-9x+14}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-9}}}, and the last term is {{{14}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{14}}} to get {{{(1)(14)=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 {{{-9}}}?



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 {{{-9}}}:



<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=15</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=9</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)=-15</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)=-9</font></td></tr></table>



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



So the two numbers {{{-2}}} and {{{-7}}} both multiply to {{{14}}} <font size=4><b>and</b></font> add to {{{-9}}}



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



{{{x^2+highlight(-2x-7x)+14}}} Replace the second term {{{-9x}}} with {{{-2x-7x}}}.



{{{(x^2-2x)+(-7x+14)}}} Group the terms into two pairs.



{{{x(x-2)+(-7x+14)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x-2)-7(x-2)}}} 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.



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



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Answer:



So {{{x^2-9x+14}}} factors to {{{(x-7)(x-2)}}}.



In other words, {{{x^2-9x+14=(x-7)(x-2)}}}.



Note: you can check the answer by expanding {{{(x-7)(x-2)}}} to get {{{x^2-9x+14}}} or by graphing the original expression and the answer (the two graphs should be identical).

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