Multiple choice

If a, b, c are odd integers, then the equation $ax^2+bx+c=0$ cannot have.

  1. Imaginary roots

  2. Real roots

  3. Irrational roots

  4. Rational roots
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D Correct answer
AI explanation

For the quadratic equation ax^2 + bx + c = 0 to have rational roots, its discriminant D = b^2 - 4ac must be a perfect square. Since a, b, and c are odd integers, b^2 is odd and 4ac is even, making the discriminant D an odd number. Additionally, D = b^2 - 4ac takes the form 8k + 1 because b^2 is 1 modulo 8 and 4ac is 0 modulo 4, but for D to be a perfect square of an odd number, it must be strictly 1 modulo 8. Wait, since b^2 - 4ac = (2n+1)^2 - 4(2p+1)(2q+1) = 4n(n+1) + 1 - 4(odd), and 4n(n+1) is a multiple of 8, D is of the form 8m + 5, which cannot be a perfect square. Therefore, the equation cannot have rational roots.

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