document.write( "Question 191092: factor by grouping\r
\n" ); document.write( "\n" ); document.write( "4x^2-4x-3 \n" ); document.write( "

Algebra.Com's Answer #143447 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"4x%5E2-4x-3\"$ , we can see that the first coefficient is $\"4\"$ , the second coefficient is $\"-4\"$ , and the last term is $\"-3\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"4\"$ by the last term $\"-3\"$ to get $\"%284%29%28-3%29=-12\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-12\"$ (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 $\"-12\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-12\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-12\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 $\"-12\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-12)
\n" ); document.write( "2*(-6)
\n" ); document.write( "3*(-4)
\n" ); document.write( "(-1)*(12)
\n" ); document.write( "(-2)*(6)
\n" ); document.write( "(-3)*(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	-12	1+(-12)=-11
2	-6	2+(-6)=-4
3	-4	3+(-4)=-1
-1	12	-1+12=11
-2	6	-2+6=4
-3	4	-3+4=1

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"2\"$ and $\"-6\"$ add to $\"-4\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"2\"$ and $\"-6\"$ both multiply to $\"-12\"$ and add to $\"-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-4x\"$ with $\"2x-6x\"$ . Remember, $\"2\"$ and $\"-6\"$ add to $\"-4\"$ . So this shows us that $\"2x-6x=-4x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"4x%5E2%2Bhighlight%282x-6x%29-3\"$ Replace the second term $\"-4x\"$ with $\"2x-6x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%284x%5E2%2B2x%29%2B%28-6x-3%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%282x%2B1%29%2B%28-6x-3%29\"$ Factor out the GCF $\"2x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%282x%2B1%29-3%282x%2B1%29\"$ 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( " $\"%282x-3%29%282x%2B1%29\"$ Combine like terms. Or factor out the common term $\"2x%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"4x%5E2-4x-3\"$ factors to $\"%282x-3%29%282x%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%282x-3%29%282x%2B1%29\"$ to get $\"4x%5E2-4x-3\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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