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Let's solve the problem step by step.

### Given:
\[
f(x) = 2kx^2 - 4kx + 1
\]
with \( k \neq 0 \). We know that \( f(x) = 0 \) has two equal roots.

### Part (a): Find the value of \( k \).

To find when a quadratic equation has two equal roots, we use the discriminant. For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant (\( \Delta \)) is:

\[
\Delta = b^2 - 4ac
\]

If the equation has two equal roots, the discriminant must be zero:

\[
\Delta = 0
\]

For the function \( f(x) = 2kx^2 - 4kx + 1 \):
- \( a = 2k \)
- \( b = -4k \)
- \( c = 1 \)

So, the discriminant becomes:

\[
\Delta = (-4k)^2 - 4(2k)(1) = 16k^2 - 8k
\]

Setting the discriminant to zero:

\[
16k^2 - 8k = 0
\]

Factor out \( 8k \):

\[
8k(2k - 1) = 0
\]

Since \( k \neq 0 \), we have:

\[
2k - 1 = 0
\]

\[
k = \frac{1}{2}
\]

### Answer for Part (a):
\[
k = \frac{1}{2}
\]

### Part (b): Find all possible values of \( p \).

We are asked where the line \( y = p \) intersects the graph of \( f(x) \). To find this, set \( f(x) = p \):

\[
2kx^2 - 4kx + 1 = p
\]

Substituting \( k = \frac{1}{2} \):

\[
2 \cdot \frac{1}{2}x^2 - 4 \cdot \frac{1}{2}x + 1 = p
\]

\[
x^2 - 2x + 1 = p
\]

\[
(x - 1)^2 = p
\]

Since \( (x - 1)^2 \) is always non-negative, \( p \) must be non-negative as well. Thus:

\[
p \geq 0
\]

### Answer for Part (b):
\[
p \geq 0
\]

Would you like further details or have any questions?

### 5 Related Questions:
1. How do you determine if a quadratic equation has no real roots using the discriminant?
2. What happens if \( k = 0 \) in the given equation?
3. Can \( p \) be a negative value in any case for this function?
4. How can we confirm if the roots of a quadratic equation are real and distinct?
5. What is the geometric interpretation of a line intersecting a parabola?

### Tip:
Remember that the discriminant determines the nature of the roots: zero indicates a double root, positive indicates two distinct roots, and negative indicates no real roots.

Consider f(x) = 2kx^2 - 4kx + 1 = 0, for k ≠ 0. The equation f(x) = 0 has two equal roots.
(a) Find the value of k.
(b) The line y = p intersects the graph of f. Find all possible values of p.

Solve the problem involving a quadratic function f(x) = 2kx^2 - 4kx + 1, where f(x) has two equal roots. Learn how to find the value of k and determine possible values for p where the line y = p intersects the graph of f. Understand the use of discriminants in quadratic equations.

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