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We are asked to find the values of \( k \) for which the quadratic function \( f(x) = x^2 - (k + 2)x - 1 \) intersects the \( x \)-axis at two points.

### Step-by-step explanation:

1. **Intersection with the x-axis:**
   A quadratic function intersects the \( x \)-axis at points where \( f(x) = 0 \), meaning we need to solve the equation:
   \[
   x^2 - (k + 2)x - 1 = 0
   \]
   This is a standard quadratic equation of the form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -(k + 2) \), and \( c = -1 \).

2. **Discriminant condition:**
   For a quadratic equation \( ax^2 + bx + c = 0 \) to have two distinct real solutions (i.e., the function intersects the x-axis at two points), the discriminant must be positive. The discriminant \( \Delta \) is given by:
   \[
   \Delta = b^2 - 4ac
   \]
   Substituting \( a = 1 \), \( b = -(k + 2) \), and \( c = -1 \) into this formula:
   \[
   \Delta = (-(k + 2))^2 - 4(1)(-1)
   \]
   Simplifying:
   \[
   \Delta = (k + 2)^2 + 4
   \]
   \[
   \Delta = k^2 + 4k + 4 + 4 = k^2 + 4k + 8
   \]

3. **Finding values of \( k \):**
   For the function to intersect the x-axis at two points, \( \Delta > 0 \). So, we need to solve:
   \[
   k^2 + 4k + 8 > 0
   \]
   Since the discriminant is always positive (the quadratic \( k^2 + 4k + 8 \) has no real roots and opens upwards), the inequality holds for all real values of \( k \).

   This means that the graph of the function intersects the \( x \)-axis at two points for **any value of \( k \)**.

### Conclusion:
The correct answer is **D) Para ningún valor de \( k \)**.

Would you like further details or have any questions?

### Related Questions:
1. What is the geometric meaning of the discriminant in a quadratic equation?
2. How does the value of \( k \) affect the shape of the parabola in this function?
3. Can the discriminant be used to determine if a quadratic equation has complex roots?
4. How would the solution change if the quadratic equation was written in a different form?
5. How can we solve quadratic equations using the quadratic formula?

**Tip:** Understanding how the discriminant relates to the roots of a quadratic equation is key to analyzing the graph's intersection with the x-axis.

¿Para qué valores de k, la gráfica de la función cuadrática f(x) = x^2 - (k + 2)x - 1 interseca en dos puntos al eje x?

Solve for the values of k where the quadratic function f(x) = x^2 - (k + 2)x - 1 intersects the x-axis at two points. This problem covers key algebraic concepts like the discriminant and quadratic functions, suitable for students in Grades 9-11.

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