Multiple choice

If a and b are odd integers, then the roots of the equation 2ax2 + (2a + b)x + b = 0, a ≠ 0, are

  1. non-real

  2. equal

  3. irrational

  4. rational
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Discriminant D = (2a+b)^2 - 4(2a)(b) = 4a^2 + 4ab + b^2 - 8ab = 4a^2 - 4ab + b^2 = (2a-b)^2. Since a and b are integers, the discriminant is a perfect square. Therefore, the roots are rational.

AI explanation

Calculate the discriminant of the equation, given by D = (2a + b)^2 - 4 * (2a) * b. This expands to 4a^2 + 4ab + b^2 - 8ab, which simplifies perfectly to the perfect square (2a - b)^2. Because a and b are integers, the discriminant is a perfect square integer, meaning the square root of the discriminant is rational. Using the quadratic formula, dividing this rational discriminant by the integer 4a results in rational roots.

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