document.write( "Question 154119This question is from textbook Intermediate Algebra
\n" ); document.write( ": Find two integers whose product is 105 such that one of the integers is one more twice the other integer. \n" ); document.write( "
Algebra.Com's Answer #113436 by nerdybill(7384)\"\" \"About 
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Find two integers whose product is 105 such that one of the integers is one more twice the other integer.
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\n" ); document.write( "Let x = one of two integers
\n" ); document.write( "and y = second of two integers
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\n" ); document.write( "Since we have two unknowns, we'll need two equations.
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\n" ); document.write( "Equation 1 comes from:\"whose product is 105 \"
\n" ); document.write( "xy = 105
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\n" ); document.write( "Equation 2 comes from:\"one of the integers is one more twice the other integer\"
\n" ); document.write( "x = 2y + 1
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\n" ); document.write( "Using the definition of 'x' from equation 2, we substitute it into equation 1 a nd solve for 'y':
\n" ); document.write( "xy = 105
\n" ); document.write( "(2y + 1)y = 105
\n" ); document.write( "2y^2 + y = 105
\n" ); document.write( "2y^2 + y - 105 = 0
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\n" ); document.write( "Since it can't easily be factored, we can use the \"quadratic equation\" to solve. Doing so yields (See reference below):
\n" ); document.write( "y = {7, -7.5}
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\n" ); document.write( "If y = 7, we can find 'x' by substituting it into equation 1 and solve for 'x':
\n" ); document.write( "xy = 105
\n" ); document.write( "x(7) = 105
\n" ); document.write( "x = 105/7
\n" ); document.write( "x = 15
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\n" ); document.write( "If y = -7.5
\n" ); document.write( "xy = 105
\n" ); document.write( "x(-7.5) = 105
\n" ); document.write( "x = 105/(-7.5)
\n" ); document.write( "x = -14
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\n" ); document.write( "Conclusion -- there are two possible answers:
\n" ); document.write( "(x,y) = (15,7)
\n" ); document.write( "(x,y) = (-14,-7.5)
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\n" ); document.write( "For reference, here is the details of the quadratic:
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Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation \"ax%5E2%2Bbx%2Bc=0\" (in our case \"2x%5E2%2B1x%2B-105+=+0\") has the following solutons:
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\n" ); document.write( " \"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"
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\n" ); document.write( " For these solutions to exist, the discriminant \"b%5E2-4ac\" should not be a negative number.
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\n" ); document.write( " First, we need to compute the discriminant \"b%5E2-4ac\": \"b%5E2-4ac=%281%29%5E2-4%2A2%2A-105=841\".
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\n" ); document.write( " Discriminant d=841 is greater than zero. That means that there are two solutions: \"+x%5B12%5D+=+%28-1%2B-sqrt%28+841+%29%29%2F2%5Ca\".
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\n" ); document.write( " \"x%5B1%5D+=+%28-%281%29%2Bsqrt%28+841+%29%29%2F2%5C2+=+7\"
\n" ); document.write( " \"x%5B2%5D+=+%28-%281%29-sqrt%28+841+%29%29%2F2%5C2+=+-7.5\"
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\n" ); document.write( " Quadratic expression \"2x%5E2%2B1x%2B-105\" can be factored:
\n" ); document.write( " \"2x%5E2%2B1x%2B-105+=+2%28x-7%29%2A%28x--7.5%29\"
\n" ); document.write( " Again, the answer is: 7, -7.5.\n" ); document.write( "Here's your graph:
\n" ); document.write( "\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B1%2Ax%2B-105+%29\"

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