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.