Answer:

To find the roots of the quadratic equation kx^2 - 2kx + 8 = 0, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

Comparing the equation with the standard form ax^2 + bx + c = 0, we have:

a = k

b = -2k

c = 8

Substituting these values into the quadratic formula, we get:

x = (-(-2k) ± √((-2k)^2 - 4(k)(8))) / (2(k))

Simplifying further:

x = (2k ± √(4k^2 - 32k)) / (2k)

x = (2k ± √(4k(k - 8))) / (2k)

x = (2k ± 2√(k(k - 8))) / (2k)

x = (k ± √(k^2 - 8k)) / k

Now, let's consider the two cases for the roots:

1. k ≠ 0:

In this case, we can divide both the numerator and denominator by k:

x = (k ± √(k^2 - 8k)) / k

x = 1 ± √(k - 8)

Therefore, the roots are:

x₁ = 1 + √(k - 8)

x₂ = 1 - √(k - 8)

2. k = 0:

If k = 0, then the equation simplifies to:

0x^2 - 2(0)x + 8 = 0

0 + 0 + 8 = 0

Since this equation has no terms involving x, there is no variable x, and hence, no roots.

In conclusion, if the value of k is 1x^2 - 2kx + 8, the roots of the quadratic equation are given by:

x₁ = 1 + √(k - 8)

x₂ = 1 - √(k - 8)