document.write( "Question 281637: Hi!\r
\n" ); document.write( "\n" ); document.write( "What is the process to convert a FOIL expression back into quadratic equation?
\n" ); document.write( "example:\r
\n" ); document.write( "\n" ); document.write( " 5p squared + 14pq -3q squared=0\r
\n" ); document.write( "\n" ); document.write( " I know that the answer is (5p-q) (p + 3q)\r
\n" ); document.write( "\n" ); document.write( " What are the operations to arrive at the above expression?\r
\n" ); document.write( "\n" ); document.write( " Thank you. \n" ); document.write( "

Algebra.Com's Answer #204611 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "Looking at $\"5p%5E2%2B14pq-3q%5E2\"$ we can see that the first term is $\"5p%5E2\"$ and the last term is $\"-3q%5E2\"$ where the coefficients are 5 and -3 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 5 and the last coefficient -3 to get -15. Now what two numbers multiply to -15 and add to the middle coefficient 14? Let's list all of the factors of -15:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -15:\r
\n" ); document.write( "\n" ); document.write( "1,3,5,15\r
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\n" ); document.write( "\n" ); document.write( "-1,-3,-5,-15 ...List the negative factors as well. 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 -15\r
\n" ); document.write( "\n" ); document.write( "(1)*(-15)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-5)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(15)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(5)\r
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\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to 14? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 14\r
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First Number	Second Number	Sum
1	-15	1+(-15)=-14
3	-5	3+(-5)=-2
-1	15	-1+15=14
-3	5	-3+5=2

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -1 and 15 add up to 14 and multiply to -15\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"5p%5E2%2B14pq-3q%5E2\"$ , replace $\"14pq\"$ with $\"-pq%2B15pq\"$ (notice $\"-pq%2B15pq\"$ adds up to $\"14pq\"$ . So it is equivalent to $\"14pq\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"5p%5E2%2Bhighlight%28-pq%2B15pq%29-3q%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"5p%5E2-pq%2B15pq-3q%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%285p%5E2-pq%29%2B%2815pq-3q%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"p%285p-q%29%2B3q%285p-q%29\"$ Factor out the GCF of $\"p\"$ out of the first group. Factor out the GCF of $\"3q\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28p%2B3q%29%285p-q%29\"$ Since we have a common term of $\"5p-q\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"5p%5E2-pq%2B15pq-3q%5E2\"$ factors to $\"%28p%2B3q%29%285p-q%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"5p%5E2%2B14pq-3q%5E2\"$ factors to $\"%28p%2B3q%29%285p-q%29\"$ (since $\"5p%5E2%2B14pq-3q%5E2\"$ is equivalent to $\"5p%5E2-pq%2B15pq-3q%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"5p%5E2%2B14pq-3q%5E2\"$ factors to $\"%28p%2B3q%29%285p-q%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"5p%5E2%2B14pq-3q%5E2=%28p%2B3q%29%285p-q%29\"$
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