Given: The quadratic equation ax^2 - bx + 5 = 0 does not have two distinct roots.

To Find: The minimum value of 5ab.

Solution:
To find the minimum value of 5ab, we need to analyze the given quadratic equation and its roots.

1. Quadratic Equation:
The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants. In this case, the quadratic equation is ax^2 - bx + 5 = 0.

2. Nature of Roots:
For a quadratic equation ax^2 + bx + c = 0, the nature of its roots is determined by the discriminant (D), given by D = b^2 - 4ac.

If the discriminant (D) is greater than 0, the equation has two distinct real roots.
If the discriminant (D) is equal to 0, the equation has two equal real roots.
If the discriminant (D) is less than 0, the equation has two complex roots.

In this case, we are given that the quadratic equation does not have two distinct roots. Therefore, the discriminant (D) must be less than 0.

3. Discriminant (D) Calculation:
For the given quadratic equation ax^2 - bx + 5 = 0, the discriminant (D) can be calculated as follows:

D = b^2 - 4ac
= (-b)^2 - 4(a)(5)
= b^2 - 20a

Since the discriminant (D) is less than 0, we can write the inequality:

b^2 - 20a < />

4. Minimum Value of 5ab:
To find the minimum value of 5ab, we need to minimize the product of a and b.

Let's consider the case where a = 1 and b = 1, which satisfies the inequality b^2 - 20a < />

5ab = 5(1)(1) = 5

Therefore, the minimum value of 5ab is 5.

Conclusion:
The minimum value of 5ab is 5, given that the quadratic equation ax^2 - bx + 5 = 0 does not have two distinct roots.