Multiple choice

The probability of choosing randomly a number c from the set ${1, 2, 3, ..........9} $ such that the quadratic equation $x^2+ 4x +c=0$ has real roots is:

  1. $\dfrac{1}{9}$
  2. $\dfrac{2}{9}$
  3. $\dfrac{3}{9}$
  4. $\dfrac{4}{9}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For x^2 + 4x + c = 0 to have real roots, the discriminant D = 16 - 4c must be >= 0. So 16 >= 4c, or c <= 4. In the set {1, 2, 3, 4, 5, 6, 7, 8, 9}, the values of c satisfying this are {1, 2, 3, 4}. There are 4 such values out of 9, so the probability is 4/9.

AI explanation

For the quadratic equation x^2+4x+c=0 to have real roots, the discriminant must be non-negative. Here, b^2-4ac is 16-4c, which must be greater than or equal to zero. This implies c is less than or equal to 4, giving the valid set of numbers as 1, 2, 3 and 4. The total numbers in the set are 9, so the required probability is 4/9.

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