Final answer:
The probability that the roots of the quadratic equation x2 + ax + 35 = 0 are real, given a is uniformly distributed over [-17, 19], is approximately 34.28%.
Step-by-step explanation:
For the quadratic equation x2 + ax + 35 = 0 to have real roots, the discriminant (b2 - 4ac) must be greater than or equal to zero. Given that a is uniformly distributed over [-17, 19], the values for a will affect the discriminant.
To find the probability of real roots, we first need to understand the discriminant in the context of our equation, which is a2 - 4(1)(35). This simplifies to a2 - 140. For the roots to be real, the discriminant must be nonnegative, which gives us a2 ≥ 140, or |a| ≥ √140. The square root of 140 is approximately 11.83, so the values of a that satisfy this are from [-17, -11.83] and [11.83, 19].
We can now find the probability using the uniform distribution. The total length of the interval of a is 19 - (-17) = 36. The length of the interval from which real roots will result is from -17 to -11.83 and from 11.83 to 19, so we add the lengths of these intervals: (17 - 11.83) + (19 - 11.83) = 5.17 + 7.17 = 12.34. Now, the probability is the length of the favorable outcomes over the length of the entire interval, which is 12.34/36 ≈ 0.3428, or 34.28%.