Question 414527


{{{-2h^2-28h-98}}} Start with the given expression.



{{{-2(h^2+14h+49)}}} Factor out the GCF {{{-2}}}.



Now let's try to factor the inner expression {{{h^2+14h+49}}}



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Looking at the expression {{{h^2+14h+49}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{14}}}, and the last term is {{{49}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{49}}} to get {{{(1)(49)=49}}}.



Now the question is: what two whole numbers multiply to {{{49}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{14}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{49}}} (the previous product).



Factors of {{{49}}}:

1,7,49

-1,-7,-49



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



These factors pair up and multiply to {{{49}}}.

1*49 = 49
7*7 = 49
(-1)*(-49) = 49
(-7)*(-7) = 49


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{14}}}:



<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>49</font></td><td  align="center"><font color=black>1+49=50</font></td></tr><tr><td  align="center"><font color=red>7</font></td><td  align="center"><font color=red>7</font></td><td  align="center"><font color=red>7+7=14</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-49</font></td><td  align="center"><font color=black>-1+(-49)=-50</font></td></tr><tr><td  align="center"><font color=black>-7</font></td><td  align="center"><font color=black>-7</font></td><td  align="center"><font color=black>-7+(-7)=-14</font></td></tr></table>



From the table, we can see that the two numbers {{{7}}} and {{{7}}} add to {{{14}}} (the middle coefficient).



So the two numbers {{{7}}} and {{{7}}} both multiply to {{{49}}} <font size=4><b>and</b></font> add to {{{14}}}



Now replace the middle term {{{14h}}} with {{{7h+7h}}}. Remember, {{{7}}} and {{{7}}} add to {{{14}}}. So this shows us that {{{7h+7h=14h}}}.



{{{h^2+highlight(7h+7h)+49}}} Replace the second term {{{14h}}} with {{{7h+7h}}}.



{{{(h^2+7h)+(7h+49)}}} Group the terms into two pairs.



{{{h(h+7)+(7h+49)}}} Factor out the GCF {{{h}}} from the first group.



{{{h(h+7)+7(h+7)}}} 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.



{{{(h+7)(h+7)}}} Combine like terms. Or factor out the common term {{{h+7}}}



{{{(h+7)^2}}} Condense the terms.



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So {{{-2(h^2+14h+49)}}} then factors further to {{{-2(h+7)^2}}}



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Answer:



So {{{-2h^2-28h-98}}} completely factors to {{{-2(h+7)^2}}}.



In other words, {{{-2h^2-28h-98=-2(h+7)^2}}}.



Note: you can check the answer by expanding {{{-2(h+7)^2}}} to get {{{-2h^2-28h-98}}} or by graphing the original expression and the answer (the two graphs should be identical).



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