If product of the real roots of the equation, `x^(2)-ax+30=2sqrt((x^(2)-ax+45)),agt0` is `lamda` minimum value of sum of roots of the equation is `mu`. The value of `(mu)` (where (.) denotes the least integer function) is

Explanation

To solve the given problem step by step, we start with the equation:

\[ x^2 - ax + 30 = 2\sqrt{x^2 - ax + 45} \]

### Step 1: Rearranging the Equation
Let \( y = x^2 - ax + 30 \). Then, we can rewrite the equation as:

\[ y = 2\sqrt{y + 15} \]

### Step 2: Squaring Both Sides
To eliminate the square root, we square both sides:

\[ y^2 = 4(y + 15) \]

### Step 3: Simplifying the Equation
Expanding the right side gives us:

\[ y^2 = 4y + 60 \]

Rearranging this leads to:

\[ y^2 - 4y - 60 = 0 \]

### Step 4: Solving the Quadratic Equation
Now we can solve the quadratic equation using the quadratic formula:

\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 1, b = -4, c = -60 \):

\[ y = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-60)}}{2 \cdot 1} \]

Calculating the discriminant:

\[ = \sqrt{16 + 240} = \sqrt{256} = 16 \]

Thus, we have:

\[ y = \frac{4 \pm 16}{2} \]

Calculating the two possible values for \( y \):

1. \( y = \frac{20}{2} = 10 \)
2. \( y = \frac{-12}{2} = -6 \) (not valid since \( y \) must be non-negative)

So, we take \( y = 10 \).

### Step 5: Substituting Back to Find \( x \)
Now, substituting back to find \( x \):

\[ x^2 - ax + 30 = 10 \]

This simplifies to:

\[ x^2 - ax + 20 = 0 \]

### Step 6: Finding the Product of Roots
For the quadratic equation \( x^2 - ax + 20 = 0 \), the product of the roots \( \alpha \beta \) is given by:

\[ \alpha \beta = \frac{c}{a} = 20 \]

According to the problem, this product is equal to \( \lambda \), so \( \lambda = 20 \).

### Step 7: Finding the Sum of Roots
The sum of the roots \( \alpha + \beta \) is given by:

\[ \alpha + \beta = a \]

### Step 8: Applying the AM-GM Inequality
Using the Arithmetic Mean-Geometric Mean (AM-GM) inequality:

\[ \frac{\alpha + \beta}{2} \geq \sqrt{\alpha \beta} \]

Substituting \( \alpha \beta = 20 \):

\[ \frac{\alpha + \beta}{2} \geq \sqrt{20} \]

Thus,

\[ \alpha + \beta \geq 2\sqrt{20} = 4\sqrt{5} \]

Calculating \( 4\sqrt{5} \):

\[ \sqrt{5} \approx 2.236 \implies 4\sqrt{5} \approx 4 \times 2.236 \approx 8.944 \]

### Step 9: Finding the Minimum Value of \( \mu \)
The minimum value of \( \alpha + \beta \) is \( 4\sqrt{5} \approx 8.944 \). Therefore, the least integer function \( \mu \) is:

\[ \mu = \lfloor 8.944 \rfloor = 8 \]

### Final Answer
Thus, the value of \( \mu \) is:

\[ \boxed{8} \]