To find the minimum value of 5ab in the given quadratic equation, we first need to determine the conditions under which the equation does not have two distinct roots.

Let's analyze the given quadratic equation: ax^2 - bx + 5 = 0.

If the quadratic equation does not have two distinct roots, it means that the discriminant (b^2 - 4ac) is either zero or negative.

The discriminant is given by b^2 - 4ac. In this case, a = a, b = -b, and c = 5. So, the discriminant becomes (-b)^2 - 4(a)(5) = b^2 - 20a.

Now, we have two cases to consider:

Case 1: Discriminant is zero (b^2 - 20a = 0)
When the discriminant is zero, the quadratic equation will have one repeated root. Therefore, it won't have two distinct roots. Substituting the discriminant equation into this case, we get:
b^2 - 20a = 0.

Case 2: Discriminant is negative (b^2 - 20a < />
When the discriminant is negative, the quadratic equation will have two complex roots. Again, this means it won't have two distinct roots. Substituting the discriminant equation into this case, we get:
b^2 - 20a < />

Now, let's solve both cases separately to find the minimum value of 5ab.

Case 1: Discriminant is zero (b^2 - 20a = 0)
Rearranging the equation, we have:
b^2 = 20a.

Since we are looking for the minimum value of 5ab, we can substitute b^2 = 20a into 5ab to get:
5ab = 5a(20a) = 100a^2.

So, the minimum value of 5ab in this case is 100a^2.

Case 2: Discriminant is negative (b^2 - 20a < />
Rearranging the inequality, we have:
b^2 < />

Since we are looking for the minimum value of 5ab, we can substitute b^2 = 20a into 5ab to get:
5ab = 5a(20a) = 100a^2.

So, the minimum value of 5ab in this case is also 100a^2.

Finally, we can conclude that the minimum value of 5ab in both cases is 100a^2.