Let's first find the discriminant of the quadratic equation:
Δ = a^2 - 4(a+24)
For the roots to be real, we need Δ to be greater than or equal to zero. Therefore, we have the inequality:
a^2 - 4(a+24) ≥ 0
Simplifying this inequality, we get:
a^2 - 4a - 96 ≥ 0
Factoring the left-hand side, we get:
(a-12)(a+8) ≥ 0
The solutions to this inequality are:
a ≤ -8 or a ≥ 12
However, we are given that a is uniformly distributed over [-17, 15]. Therefore, the probability of a being less than or equal to -8 or greater than or equal to 12 is:
P(a ≤ -8 or a ≥ 12) = P(a ≤ -8) + P(a ≥ 12)
Since a is uniformly distributed over [-17, 15], we have:
P(a ≤ -8) = (|-8 - (-17)|) / (|15 - (-17)|) = 9/32
P(a ≥ 12) = (|15 - 12|) / (|15 - (-17)|) = 3/32
Therefore, the probability that the roots of the equation x^2+ax+a+24=0 are both real is:
P(a ≤ -8 or a ≥ 12) = 9/32 + 3/32 = 3/8