Question 601670


Looking at the expression {{{h^2+10h+25}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{10}}}, and the last term is {{{25}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{25}}} to get {{{(1)(25)=25}}}.



Now the question is: what two whole numbers multiply to {{{25}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{10}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{25}}} (the previous product).



Factors of {{{25}}}:

1,5,25

-1,-5,-25



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



These factors pair up and multiply to {{{25}}}.

1*25 = 25
5*5 = 25
(-1)*(-25) = 25
(-5)*(-5) = 25


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{10}}}:



<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>25</font></td><td  align="center"><font color=black>1+25=26</font></td></tr><tr><td  align="center"><font color=red>5</font></td><td  align="center"><font color=red>5</font></td><td  align="center"><font color=red>5+5=10</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-25</font></td><td  align="center"><font color=black>-1+(-25)=-26</font></td></tr><tr><td  align="center"><font color=black>-5</font></td><td  align="center"><font color=black>-5</font></td><td  align="center"><font color=black>-5+(-5)=-10</font></td></tr></table>



From the table, we can see that the two numbers {{{5}}} and {{{5}}} add to {{{10}}} (the middle coefficient).



So the two numbers {{{5}}} and {{{5}}} both multiply to {{{25}}} <font size=4><b>and</b></font> add to {{{10}}}



Now replace the middle term {{{10h}}} with {{{5h+5h}}}. Remember, {{{5}}} and {{{5}}} add to {{{10}}}. So this shows us that {{{5h+5h=10h}}}.



{{{h^2+highlight(5h+5h)+25}}} Replace the second term {{{10h}}} with {{{5h+5h}}}.



{{{(h^2+5h)+(5h+25)}}} Group the terms into two pairs.



{{{h(h+5)+(5h+25)}}} Factor out the GCF {{{h}}} from the first group.



{{{h(h+5)+5(h+5)}}} Factor out {{{5}}} 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+5)(h+5)}}} Combine like terms. Or factor out the common term {{{h+5}}}



{{{(h+5)^2}}} Condense the terms.



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Answer:



So {{{h^2+10h+25}}} factors to {{{(h+5)^2}}}.



In other words, {{{h^2+10h+25=(h+5)^2}}}.



Note: you can check the answer by expanding {{{(h+5)^2}}} to get {{{h^2+10h+25}}} or by graphing the original expression and the answer (the two graphs should be identical).


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