document.write( "Question 694634: 9x^2-36x+36 factored \n" ); document.write( "

Algebra.Com's Answer #428098 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"9x%5E2-36x%2B36\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"9%28x%5E2-4x%2B4%29\"$ Factor out the GCF $\"9\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"x%5E2-4x%2B4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2-4x%2B4\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-4\"$ , and the last term is $\"4\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"4\"$ to get $\"%281%29%284%29=4\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"4\"$ (the previous product) and add to the second coefficient $\"-4\"$ ?\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 $\"4\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"4\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4\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 $\"4\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*4 = 4
\n" ); document.write( "2*2 = 4
\n" ); document.write( "(-1)*(-4) = 4
\n" ); document.write( "(-2)*(-2) = 4\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 $\"-4\"$ :\r
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First Number	Second Number	Sum
1	4	1+4=5
2	2	2+2=4
-1	-4	-1+(-4)=-5
-2	-2	-2+(-2)=-4

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-2\"$ and $\"-2\"$ add to $\"-4\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-2\"$ and $\"-2\"$ both multiply to $\"4\"$ and add to $\"-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-4x\"$ with $\"-2x-2x\"$ . Remember, $\"-2\"$ and $\"-2\"$ add to $\"-4\"$ . So this shows us that $\"-2x-2x=-4x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%28-2x-2x%29%2B4\"$ Replace the second term $\"-4x\"$ with $\"-2x-2x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-2x%29%2B%28-2x%2B4%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-2%29%2B%28-2x%2B4%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-2%29-2%28x-2%29\"$ Factor out $\"2\"$ 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( " $\"%28x-2%29%28x-2%29\"$ Combine like terms. Or factor out the common term $\"x-2\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28x-2%29%5E2\"$ Condense the terms.\r
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\n" ); document.write( "\n" ); document.write( "So $\"9%28x%5E2-4x%2B4%29\"$ then factors further to $\"9%28x-2%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"9x%5E2-36x%2B36\"$ completely factors to $\"9%28x-2%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"9x%5E2-36x%2B36=9%28x-2%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"9%28x-2%29%5E2\"$ to get $\"9x%5E2-36x%2B36\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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