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Let n = first integer
\n" ); document.write( "Then the second and third consecutive integers are n+1 and n+2, respectively.
\n" ); document.write( "Your problem statement gives us
\n" ); document.write( "(1) n*(n+1)*(n+2) = n^3 - 21, agree?
\n" ); document.write( "Simplify (1) to get
\n" ); document.write( "(2) n*(n^2 +3n +2) = n^3 + 21 or
\n" ); document.write( "(3) n^3 + 3n^2 + 2n = n^3 + 21
\n" ); document.write( "The n^3 terms of (3) cancel leaving
\n" ); document.write( "(4) 3n^2 + 2n - 21 = 0
\n" ); document.write( "Are you good at factoring? No, use the quadratic factoring equation.
\n" ); document.write( "I'm fair enough at factoring to be able to use the basic \"rules\" and get
\n" ); document.write( "(5) 3n^2 + 2n -21 = (3n - x)*(n + y), where x*y = 21, try 7*3 and get
\n" ); document.write( "(6) 3n^2 + 2n -21 = (3n-7)*(n+3)
\n" ); document.write( "If you FOIL the right side of (6) you wili get the left side of (6).
\n" ); document.write( "Setting (6) equal to zero you get the two roots of (4) as
\n" ); document.write( "(7) n = (7/3,-3).
\n" ); document.write( "The first root, 7/3 is not a solution because 7/3 is not an integer. The correct root is
\n" ); document.write( "(8) n = -3
\n" ); document.write( "Let's check the value n = -3 with (1).
\n" ); document.write( "Is (-3*-2*-1 = (-3)^3 + 21)?
\n" ); document.write( "Is (-6 = -27 +21)?
\n" ); document.write( "Is (-6 = -6)? Yes
\n" ); document.write( "Answer: The three consecutive integers are -3, -2, and -1. \n" ); document.write( "