Multiple choice

If $p$ is chosen at random in the closed interval $[0,5]$. The probability of the equation $\displaystyle { x }^{ 2 }+px+\frac { 1 }{ 4 } \left( p+2 \right) =0$ to have the real roots is

  1. $\displaystyle \frac { 3 }{ 4 } $
  2. $\displaystyle \frac { 3 }{ 5 } $
  3. $\displaystyle \frac { 2 }{ 5 } $

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For the quadratic equation to have real roots, the discriminant D = p^2 - 4(1/4)(p+2) must be >= 0. This simplifies to p^2 - p - 2 >= 0, or (p-2)(p+1) >= 0. Given p is in [0, 5], the condition holds for p in [2, 5]. The probability is the length of this interval (5-2=3) divided by the total interval length (5-0=5), which is 3/5.

AI explanation

For the equation x^2 + px + (1/4)(p + 2) = 0 to have real roots, the discriminant must be non-negative, giving the inequality p^2 - p - 2 >= 0. Factoring this quadratic inequality gives (p - 2)(p + 1) >= 0, which means p must be less than or equal to -1 or greater than or equal to 2. Within the given interval [0, 5], the valid range for p is [2, 5], which has a length of 3. The probability is the length of the valid interval divided by the total length of 5, giving 3/5.

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