Multiple choice

If coefficients $a,b,c$ of quadratic equation $a{ x }^{ 2 }+bx+c=0$ are chosen at random with replacement from the set $S={1,2,3,4,5,6}$, find out the probability that roots of quadratic are real and distinct.

  1. $\cfrac{20}{108}$
  2. $\cfrac{19}{108}$
  3. $\cfrac{21}{108}$
  4. $\cfrac{22}{108}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For the roots to be real and distinct, the discriminant D = b^2 - 4ac must be greater than 0. By testing all 216 possible combinations (6*6*6), there are 19 cases where b^2 > 4ac.

AI explanation

For the roots of ax^2 + bx + c = 0 to be real and distinct, the discriminant b^2 - 4ac must be greater than zero, meaning b^2 > 4ac. Since a and c are positive integers between 1 and 6, we count the combinations where this inequality holds, which are 19 cases: 1 way for a=1, 3 ways for a=2, 6 ways for a=3, 5 ways for a=4, 3 ways for a=5, and 1 way for a=6. With 6 possible outcomes for each of the three die rolls, there are 216 total possible outcomes. The probability is the number of favorable outcomes divided by the total outcomes, resulting in 19/216, which is equivalent to 19/108.

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