Multiple choice

Determine the number of negative integral values of h for which the equation 3x2 - 25x - h = 0 has real roots.

  1. 51

  2. 52

  3. 53

  4. Cannot be determined

  5. No real roots exist.
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B Correct answer
Explanation

For real roots, the discriminant D = b^2 - 4ac >= 0. Here, (-25)^2 - 4(3)(-h) >= 0, so 625 + 12h >= 0. 12h >= -625, so h >= -52.08. Since h must be a negative integer, h can be any integer from -52 to -1, totaling 52 values.

AI explanation

For the equation to have real roots, the discriminant must be greater than or equal to zero, so (-25)^2 - 4(3)(-h) >= 0. This simplifies to 625 + 12h >= 0, which means h >= -625/12 or h >= -52.08. The negative integers satisfying this condition are from -52 to -1, giving exactly 52 negative integral values.

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