Multiple choice

How many possible values are there for m (a non-negative integer) such that the quadratic equation 4x2 + 16x + 4m = 0 has real roots?

  1. 4

  2. 5

  3. 6

  4. 3

  5. 1
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B Correct answer
Explanation

For 4x^2 + 16x + 4m = 0 to have real roots, the discriminant D = 16^2 - 4(4)(4m) >= 0. 256 - 64m >= 0 -> 64m <= 256 -> m <= 4. Since m is a non-negative integer, m can be 0, 1, 2, 3, 4. There are 5 possible values.

AI explanation

For the quadratic equation 4x^2 + 16x + 4m = 0 to have real roots, the discriminant must be greater than or equal to zero, so b^2 - 4ac >= 0. Substituting the values gives 16^2 - 4(4)(4m) >= 0, which simplifies to 256 - 64m >= 0, or 4 >= m. Since m is a non-negative integer, the possible values are 0, 1, 2, 3, and 4, giving a total of 5 possible values.

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