document.write( "Question 452716: Factor x^2+6x+9 \n" ); document.write( "

Algebra.Com's Answer #311179 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2%2B6x%2B9\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"6\"$ , and the last term is $\"9\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"9\"$ to get $\"%281%29%289%29=9\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"9\"$ (the previous product) and add to the second coefficient $\"6\"$ ?\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 $\"9\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"9\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,3,9\r
\n" ); document.write( "\n" ); document.write( "-1,-3,-9\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 $\"9\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*9 = 9
\n" ); document.write( "3*3 = 9
\n" ); document.write( "(-1)*(-9) = 9
\n" ); document.write( "(-3)*(-3) = 9\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 $\"6\"$ :\r
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First Number	Second Number	Sum
1	9	1+9=10
3	3	3+3=6
-1	-9	-1+(-9)=-10
-3	-3	-3+(-3)=-6

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"3\"$ and $\"3\"$ add to $\"6\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"3\"$ and $\"3\"$ both multiply to $\"9\"$ and add to $\"6\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"6x\"$ with $\"3x%2B3x\"$ . Remember, $\"3\"$ and $\"3\"$ add to $\"6\"$ . So this shows us that $\"3x%2B3x=6x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%283x%2B3x%29%2B9\"$ Replace the second term $\"6x\"$ with $\"3x%2B3x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B3x%29%2B%283x%2B9%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B3%29%2B%283x%2B9%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B3%29%2B3%28x%2B3%29\"$ Factor out $\"3\"$ 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( " $\"%28x%2B3%29%28x%2B3%29\"$ Combine like terms. Or factor out the common term $\"x%2B3\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B3%29%5E2\"$ 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 $\"x%5E2%2B6x%2B9\"$ factors to $\"%28x%2B3%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"x%5E2%2B6x%2B9=%28x%2B3%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x%2B3%29%5E2\"$ to get $\"x%5E2%2B6x%2B9\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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