Multiple choice

If p is chosen at random in the interval 0 ≤ p ≤ 5, the probability that the roots of the equation x2 + px + p/4 + 1/2 = 0 are real is

  1. 1/5

  2. 2/5

  3. 4/5

  4. 3/5
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For real roots, D = p^2 - 4(p/4 + 1/2) >= 0. p^2 - p - 2 >= 0. (p-2)(p+1) >= 0. Since p is in [0, 5], the condition holds for p in [2, 5]. Length of interval = 5 - 2 = 3. Total interval length = 5. Probability = 3/5.

AI explanation

For the roots of x2 + px + p/4 + 1/2 = 0 to be real, the discriminant (D = b2 - 4ac) must be non-negative, giving p2 - 4(p/4 + 1/2) >= 0. Simplifying this inequality yields p2 - p - 2 >= 0, which factors as (p - 2)(p + 1) >= 0. This condition is met when p <= -1 or p >= 2; since p is chosen from the interval 0 to 5, the valid range is 5 - 2 = 3. The probability is the valid length divided by the total interval length (3/5), making the probability 3/5.

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