document.write( "Question 1017072: If a, b, c are odd integers, then the equation $\"+ax%5E2%2Bbx%2Bc=0+\"$ has no FRACTION solution. \r
\n" ); document.write( "\n" ); document.write( "I know how to prove, if it said integer solution, but to prove that theres no fraction solution, it seems a little hard for me. \r
\n" ); document.write( "\n" ); document.write( "Im really sorry if its the wrong section\r
\n" ); document.write( "\n" ); document.write( "Thanks~! \n" ); document.write( "

Algebra.Com's Answer #633409 by richard1234(7193) $\"\"$ $\"About$

You can put this solution on YOUR website!
You probably mean *rational* solution (since \"fraction\" is somewhat ambiguous).\r
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\n" ); document.write( "\n" ); document.write( "The roots of the equation are

. It suffices to prove that

cannot be a perfect square, otherwise the solutions would be rational.\r
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\n" ); document.write( "\n" ); document.write( "If

is a perfect square, then

where a, b, c are odd integers and n is an integer. Note that n is odd, since b^2 is odd and 4ac is even.\r
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\n" ); document.write( "\n" ); document.write( "Here, we use a little modular arithmetic. The right hand side must leave a remainder of 4 when divided by 8, since a and c are odd. However, all of the odd squares leave a remainder of 1 when divided by 8, meaning that their difference is a multiple of 8. Since the remainders upon division by 8 are not equal, the expressions

and

cannot possibly be equal, and there is no solution in odd integers a,b,c. Therefore

cannot be a perfect square, no rational solution.\r
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\n" ); document.write( "\n" ); document.write( "In modular arithmetic terms, we say that

and

. \n" ); document.write( "

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