document.write( "Question 1200102: Find three consecutive even integers such that the product of the first and second is 20 less than 10 times the third \n" ); document.write( "
Algebra.Com's Answer #834120 by math_tutor2020(3817)\"\" \"About 
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\n" ); document.write( "Consecutive integers follow one after another.
\n" ); document.write( "Examples:
\n" ); document.write( "6,7,8
\n" ); document.write( "22,23,24\r
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\n" ); document.write( "\n" ); document.write( "Consecutive even integers are when all of the values are even, and they are as close as possible.
\n" ); document.write( "Examples:
\n" ); document.write( "2,4,6
\n" ); document.write( "100,102,104\r
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\n" ); document.write( "\n" ); document.write( "Let x be some integer. It might be even or odd.
\n" ); document.write( "To guarantee it is always even, we double it to 2x.
\n" ); document.write( "Eg:
\n" ); document.write( "x = 5 doubles to 2x = 2*5 = 10
\n" ); document.write( "x = 6 doubles to 2x = 2*6 = 12\r
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\n" ); document.write( "\n" ); document.write( "2x = first even integer
\n" ); document.write( "2x+2 = even integer right after 2x
\n" ); document.write( "2x+2+2 = 2x+4 = even integer right after 2x+2\r
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\n" ); document.write( "\n" ); document.write( "In short,
\n" ); document.write( "first = 2x
\n" ); document.write( "second = 2x+2
\n" ); document.write( "third = 2x+4\r
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\n" ); document.write( "\n" ); document.write( "The sequence {2x,2x+2,2x+4} represents a set of three consecutive even integers, where x is any integer.\r
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\n" ); document.write( "\n" ); document.write( "We then have the following information \" the product of the first and second is 20 less than 10 times the third\".\r
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\n" ); document.write( "\n" ); document.write( "Let's translate that into an equation and solve for x.\r
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\n" ); document.write( "\n" ); document.write( "first*second = 10*third - 20
\n" ); document.write( "2x*(2x+2) = 10*(2x+4) - 20
\n" ); document.write( "4x^2+4x = 20x+40 - 20
\n" ); document.write( "4x^2+4x = 20x+20
\n" ); document.write( "4x^2+4x-20x-20 = 0
\n" ); document.write( "4x^2-16x-20 = 0
\n" ); document.write( "4(x^2-4x-5) = 0
\n" ); document.write( "x^2-4x-5 = 0
\n" ); document.write( "(x-5)(x+1) = 0
\n" ); document.write( "x-5 = 0 or x+1 = 0
\n" ); document.write( "x = 5 or x = -1\r
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\n" ); document.write( "\n" ); document.write( "Optionally you could use the quadratic formula to solve for x.
\n" ); document.write( "Let me know if you need to see these steps.\r
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\n" ); document.write( "\n" ); document.write( "If x = 5, then,
  • 2x = 2*5 = 10 is the first even integer
  • 2x+2 = 2*5+2 = 10+2 = 12 is the second even integer
  • 2x+4 = 2*5+4 = 10+4 = 14 is the third even integer
We end up with the set {10,12,14} as one solution.\r
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\n" ); document.write( "\n" ); document.write( "If x = -1, then,
  • 2x = 2*(-1) = -2 is the first even integer
  • 2x+2 = 2*(-1)+2 = -2+2 = 0 is the second even integer
  • 2x+4 = 2*(-1)+4 = -2+4 = 2 is the third even integer
We end up with the set {-2,0,2} as the other solution.\r
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\n" ); document.write( "\n" ); document.write( "Check for the first solution:
\n" ); document.write( "first*second = 10*third - 20
\n" ); document.write( "10*12 = 10*14 - 20
\n" ); document.write( "120 = 140 - 20
\n" ); document.write( "120 = 120
\n" ); document.write( "The first solution set is confirmed.\r
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\n" ); document.write( "\n" ); document.write( "Check for the second solution:
\n" ); document.write( "first*second = 10*third - 20
\n" ); document.write( "-2*0 = 10*2 - 20
\n" ); document.write( "0 = 20 - 20
\n" ); document.write( "0 = 0
\n" ); document.write( "The second solution set is confirmed.
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