The given equation is (x-1)(x-5)k = 0.

To find the value of k, we need to determine the roots of the equation and analyze their properties.

Finding the Roots:
The equation (x-1)(x-5)k = 0 can be rewritten as (x-1)(x-5) = 0 since k ≠ 0.

Expanding the equation, we get x^2 - 6x + 5 = 0.

To solve this quadratic equation, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

Here, a = 1, b = -6, and c = 5.

Substituting the values, we get:

x = (-(-6) ± √((-6)^2 - 4(1)(5))) / (2(1))
= (6 ± √(36 - 20)) / 2
= (6 ± √16) / 2
= (6 ± 4) / 2

Simplifying further, we have two possible values for x:

x1 = (6 + 4) / 2 = 10 / 2 = 5
x2 = (6 - 4) / 2 = 2 / 2 = 1

Therefore, the roots of the equation are x = 5 and x = 1.

Difference between the Roots:
The difference between two numbers is calculated by subtracting the smaller number from the larger number. In this case, the larger root is 5 and the smaller root is 1.

Difference = 5 - 1 = 4

Given that the difference between the roots is 2, we can set up the following equation:

5 - 1 = 2

Simplifying, we get:

4 = 2

This is not a true statement, which means that our assumption about the difference between the roots is incorrect.

Determining the Value of k:
Since the difference between the roots is not 2, we need to find an alternative approach to determine the value of k.

Let's consider the equation (x-1)(x-5)k = 0. If this equation is satisfied, it means that either (x-1) = 0 or (x-5) = 0.

If (x-1) = 0, then x = 1.
If (x-5) = 0, then x = 5.

Substituting these values into the equation, we get:

(1-1)(1-5)k = 0
0 * (-4)k = 0
0 = 0

(5-1)(5-5)k = 0
4 * 0k = 0
0 = 0

In both cases, the equation is satisfied. This means that for any value of k, the given equation will always be true. Therefore, the value of k can be any real number except 0.

In conclusion, the value of k is any real number except 0.

I am a medical student not non medical