document.write( "Question 282643: x^2 + 30x + 225. Factor completly. \n" ); document.write( "

Algebra.Com's Answer #205209 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2%2B30x%2B225\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"30\"$ , and the last term is $\"225\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"225\"$ to get $\"%281%29%28225%29=225\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"225\"$ (the previous product) and add to the second coefficient $\"30\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"225\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"225\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,3,5,9,15,25,45,75,225\r
\n" ); document.write( "\n" ); document.write( "-1,-3,-5,-9,-15,-25,-45,-75,-225\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"225\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*225 = 225
\n" ); document.write( "3*75 = 225
\n" ); document.write( "5*45 = 225
\n" ); document.write( "9*25 = 225
\n" ); document.write( "15*15 = 225
\n" ); document.write( "(-1)*(-225) = 225
\n" ); document.write( "(-3)*(-75) = 225
\n" ); document.write( "(-5)*(-45) = 225
\n" ); document.write( "(-9)*(-25) = 225
\n" ); document.write( "(-15)*(-15) = 225\r
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\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"30\"$ :\r
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First Number	Second Number	Sum
1	225	1+225=226
3	75	3+75=78
5	45	5+45=50
9	25	9+25=34
15	15	15+15=30
-1	-225	-1+(-225)=-226
-3	-75	-3+(-75)=-78
-5	-45	-5+(-45)=-50
-9	-25	-9+(-25)=-34
-15	-15	-15+(-15)=-30

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"15\"$ and $\"15\"$ add to $\"30\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"15\"$ and $\"15\"$ both multiply to $\"225\"$ and add to $\"30\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"30x\"$ with $\"15x%2B15x\"$ . Remember, $\"15\"$ and $\"15\"$ add to $\"30\"$ . So this shows us that $\"15x%2B15x=30x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%2815x%2B15x%29%2B225\"$ Replace the second term $\"30x\"$ with $\"15x%2B15x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B15x%29%2B%2815x%2B225%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B15%29%2B%2815x%2B225%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B15%29%2B15%28x%2B15%29\"$ Factor out $\"15\"$ 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.\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B15%29%28x%2B15%29\"$ Combine like terms. Or factor out the common term $\"x%2B15\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B15%29%5E2\"$ Condense the terms.\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B30x%2B225\"$ factors to $\"%28x%2B15%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"x%5E2%2B30x%2B225=%28x%2B15%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x%2B15%29%5E2\"$ to get $\"x%5E2%2B30x%2B225\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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