document.write( "Question 1027170: Given odd integers a, b, c, prove that the equation $\"ax%5E2%2Bbx%2Bc=0\"$ cannot have a solution x which is a rational number. \n" ); document.write( "

Algebra.Com's Answer #642425 by richard1234(7193) $\"\"$ $\"About$

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Assume otherwise that

has a rational solution

. Then

is rational if and only if the discriminant

is a perfect square (using the quadratic formula, coupled with the fact that a,b,c are integers).\r
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\n" ); document.write( "\n" ); document.write( "a,b,c are odd. We will show that

cannot possibly be a perfect square by looking at it modulo 8. All of the odd perfect squares (1^2, 3^2, 5^2, 7^2) leave a remainder of 1 when divided by 8. However

and

, so

, i.e. b^2 - 4ac always leaves a remainder of 5 when divided by 8. Therefore it cannot be a perfect square, and any real solution x cannot be rational.\r
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