Multiple choice

If $p, q$ are odd integers, then the roots of the equation $2px^{2} + (2p + q) x + q = 0$ are

  1. Rational

  2. Irrational

  3. Non-real

  4. Equal
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A Correct answer
Explanation

The discriminant of the quadratic equation 2px^2 + (2p + q)x + q = 0 is D = (2p + q)^2 - 4(2p)(q) = 4p^2 + 4pq + q^2 - 8pq = 4p^2 - 4pq + q^2 = (2p - q)^2. Since p and q are integers, the discriminant is a perfect square, which implies the roots are rational.

AI explanation

The discriminant of the equation 2px^2 + (2p + q)x + q = 0 is D = (2p + q)^2 - 4(2p)(q), which simplifies to 4p^2 + q^2. Since p and q are odd integers, their squares are also odd integers, making D a sum of two odd integers, which is even and positive. Therefore, the roots are real and rational because the discriminant is a perfect square or rational, making the roots rational.

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