document.write( "Question 856303: Find two consecutive integers such that three Times the square if the first integer plus three is equal to 5 times the second integer.
\n" ); document.write( "Can someone explain how to solve this please. \n" ); document.write( "

Algebra.Com's Answer #516187 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
First, you set up an equation and solve it.
\n" ); document.write( "Then, you discard any solution that does not fit the description.
\n" ); document.write( "(In this case the description is \"integer\",
\n" ); document.write( "but some problems refer to \"even integer\", or \"odd integer\").
\n" ); document.write( "
\n" ); document.write( "You need to define your \"two consecutive integers\".
\n" ); document.write( "You may write $\"n\"$ = the first integer,
\n" ); document.write( "to define your variable.
\n" ); document.write( "Or you could also use $\"x\"$ for your variable.
\n" ); document.write( "
\n" ); document.write( "Using $\"x\"$ , you know (and may, or may not want to write) that:
\n" ); document.write( " $\"x%2B1\"$ = the second integer
\n" ); document.write( " $\"x%5E2\"$ = the square of the first integer,
\n" ); document.write( " $\"3x%5E2\"$ = three times the square of the first integer,
\n" ); document.write( " $\"3x%5E2%2B3\"$ = three times the square of the first integer plus three, and
\n" ); document.write( " $\"5%28x%2B1%29\"$ = 5 times the second integer
\n" ); document.write( "
\n" ); document.write( "Your equation is
\n" ); document.write( " $\"3x%5E2%2B3=5%28x%2B1%29\"$
\n" ); document.write( "
\n" ); document.write( "Then you work on rearranging and simplifying your equation as needed:
\n" ); document.write( " $\"3x%5E2%2B3=5%28x%2B1%29\"$
\n" ); document.write( " $\"3x%5E2%2B3=5x%2B5\"$
\n" ); document.write( " $\"3x%5E2%2B3-5x-5=0\"$
\n" ); document.write( " $\"3x%5E2-5x-2=0\"$
\n" ); document.write( "
\n" ); document.write( "Finally, you solve by a method of your choice.
\n" ); document.write( "In this case, I would use the quadratic formula,
\n" ); document.write( "which says that the solutions to
\n" ); document.write( " $\"ax%5E2%2Bbx%2Bc=0\"$ are given by
\n" ); document.write( " $\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"$ .
\n" ); document.write( "In this case, applying the quadratic formula, we get
\n" ); document.write( " $\"x+=+%28-%28-5%29+%2B-+sqrt%28%28-5%29%5E2-4%2A3%2A%28-2%29+%29%29%2F%282%2A3%29+\"$
\n" ); document.write( " $\"x+=+%285+%2B-+sqrt%2825%2B24+%29%29%2F6\"$
\n" ); document.write( " $\"x+=+%285+%2B-+sqrt%2849%29%29%2F6\"$
\n" ); document.write( " $\"n+=+%285+%2B-+7%29%2F6\"$
\n" ); document.write( "You get the solutions
\n" ); document.write( " $\"x+=+%285+%2B7%29%2F6\"$ --> $\"x=12%2F6\"$ --> $\"highlight%28x=2%29\"$
\n" ); document.write( "and $\"x+=+%285+-7%29%2F6\"$ --> $\"x=-2%2F6\"$ --> $\"x=-1%2F3\"$
\n" ); document.write( "Since $\"-1%2F3\"$ is not an integer, it is not a solution to the problem, and you discard it.
\n" ); document.write( "The two consecutive integers are $\"x=highlight%282%29\"$ and $\"x%2B1=2%2B1=highlight%283%29\"$ .
\n" ); document.write( "
\n" ); document.write( "Verification:
\n" ); document.write( " $\"3%2A2%5E2%2B3=3%2A4%2B3=12%2B3=15\"$ would be three times the square of the first integer plus three;
\n" ); document.write( " $\"5%2A3=15\"$ would be \"5 times the second integer\",
\n" ); document.write( "and both quantities are equal, as the problem required. \n" ); document.write( "

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