document.write( "Question 1132277: two positive integers have a sum 0f 17 and a product of 66. What are the integers? \n" ); document.write( "
Algebra.Com's Answer #749257 by Alan3354(69443)\"\" \"About 
You can put this solution on YOUR website!
Try pairs of factors of 66.
\n" ); document.write( "Should take less than 10 seconds to find them.
\n" ); document.write( "===================
\n" ); document.write( "Copied from the other tutor's solution:
\n" ); document.write( "------------
\n" ); document.write( "------------
\n" ); document.write( "1) i +j = 17
\n" ); document.write( ":
\n" ); document.write( "2) ij = 66
\n" ); document.write( ":
\n" ); document.write( "solve equation 1 for i
\n" ); document.write( ":
\n" ); document.write( "i = 17 -j
\n" ); document.write( ":
\n" ); document.write( "substitute for i in equation 2
\n" ); document.write( ":
\n" ); document.write( "(17 -j)j = 66
\n" ); document.write( ":
\n" ); document.write( "17j -j^2 = 66
\n" ); document.write( ":
\n" ); document.write( "j^2 -17j +66 = 0
\n" ); document.write( "At this point, you find a pair of factors of 66 with a sum of 17. Similar to \"going around the barn to get to the door.\"
\n" ); document.write( "--------
\n" ); document.write( "But, if they're not integers, eg, the sum is 17 and the product is 65:
\n" ); document.write( "---
\n" ); document.write( "1) i +j = 17
\n" ); document.write( ":
\n" ); document.write( "2) ij = 65
\n" ); document.write( ":
\n" ); document.write( "i = 17 -j
\n" ); document.write( ":
\n" ); document.write( "j^2 -17j +65 = 0
\n" ); document.write( ":
\n" ); document.write( "(17 -j)j = 65
\n" ); document.write( ":
\n" ); document.write( "17j -j^2 = 65
\n" ); document.write( "j^2 - 17j + 65 = 0
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "\n" ); document.write( "
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation \"ax%5E2%2Bbx%2Bc=0\" (in our case \"1x%5E2%2B-17x%2B65+=+0\") has the following solutons:
\n" ); document.write( "
\n" ); document.write( " \"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"
\n" ); document.write( "
\n" ); document.write( " For these solutions to exist, the discriminant \"b%5E2-4ac\" should not be a negative number.
\n" ); document.write( "
\n" ); document.write( " First, we need to compute the discriminant \"b%5E2-4ac\": \"b%5E2-4ac=%28-17%29%5E2-4%2A1%2A65=29\".
\n" ); document.write( "
\n" ); document.write( " Discriminant d=29 is greater than zero. That means that there are two solutions: \"+x%5B12%5D+=+%28--17%2B-sqrt%28+29+%29%29%2F2%5Ca\".
\n" ); document.write( "
\n" ); document.write( " \"x%5B1%5D+=+%28-%28-17%29%2Bsqrt%28+29+%29%29%2F2%5C1+=+11.1925824035673\"
\n" ); document.write( " \"x%5B2%5D+=+%28-%28-17%29-sqrt%28+29+%29%29%2F2%5C1+=+5.80741759643275\"
\n" ); document.write( "
\n" ); document.write( " Quadratic expression \"1x%5E2%2B-17x%2B65\" can be factored:
\n" ); document.write( " \"1x%5E2%2B-17x%2B65+=+%28x-11.1925824035673%29%2A%28x-5.80741759643275%29\"
\n" ); document.write( " Again, the answer is: 11.1925824035673, 5.80741759643275.\n" ); document.write( "Here's your graph:
\n" ); document.write( "\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-17%2Ax%2B65+%29\"

\n" ); document.write( "\n" ); document.write( "-----
\n" ); document.write( "j =~ 11.19 or 5.8
\n" ); document.write( " \n" ); document.write( "
\n" );