document.write( "Question 793182: If A and B are positive integers such that both 4x2 +Ax+36 and 6x2+Bx+24 factor over the integers, find the largest possible value of A-B. \n" ); document.write( "

Algebra.Com's Answer #480278 by KMST(5328) $\"\"$ $\"About$

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If $\"4x%5E2%2BAx%2B36\"$ can be factored, to find the factors I would start by looking for pairs of factors of $\"4%2A36=144\"$
\n" ); document.write( "With $\"A%3E0\"$ , A would be the sum of one of those factor pairs.
\n" ); document.write( "Here are the factor pairs and their sums:
\n" ); document.write( " $\"1%2A144=144\"$ $\"1%2B144=145\"$
\n" ); document.write( " $\"2%2A72=144\"$ $\"2%2B72=74\"$
\n" ); document.write( " $\"3%2A48=144\"$ $\"3%2B48=51\"$
\n" ); document.write( " $\"4%2A36=144\"$ $\"4%2B36=40\"$
\n" ); document.write( " $\"6%2A24=144\"$ $\"6%2B24=30\"$
\n" ); document.write( " $\"8%2A18=144\"$ $\"8%2B18=26\"$
\n" ); document.write( " $\"12%2A12=144\"$ $\"12%2B12=24\"$
\n" ); document.write( "
\n" ); document.write( "If $\"6x%5E2%2BBx%2B24\"$ can be factored, to find the factors I would start by looking for pairs of factors of $\"6%2A24=144\"$
\n" ); document.write( "With $\"B%3E0\"$ , B would be the sum of one of those factor pairs, which were already listed above.
\n" ); document.write( "
\n" ); document.write( "To get the largest possible value of A-B, I need to pair possible value of A with the smallest possible value of B.
\n" ); document.write( "So the largest difference would be $\"145-24=highlight%28121%29\"$
\n" ); document.write( "
\n" ); document.write( "In that case the factorings would be
\n" ); document.write( " $\"4x%5E2%2B145x%2B36=%284x%2B1%29%28x%2B36%29\"$ and $\"6x%5E2%2B24x%2B24=6%28x%2B2%29%5E2\"$ \n" ); document.write( "

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