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The image provided contains a mathematical problem. The text reads:

"Find the values of \( k \) for which the x-axis is a tangent to the curve \( y = x^2 + (2k)x - (2k + 1) \)."

### Solution:

To determine when the x-axis is a tangent to the curve, we need to analyze the condition under which the curve touches the x-axis. 

The curve touches the x-axis if the quadratic equation \( y = 0 \) has exactly one root. For the given equation:

\[
y = x^2 + (2k)x - (2k + 1)
\]

The corresponding quadratic equation is:

\[
x^2 + (2k)x - (2k + 1) = 0
\]

For this quadratic equation to have exactly one root, the discriminant must be zero.

The discriminant \(\Delta\) for a quadratic equation of the form \(ax^2 + bx + c = 0\) is given by:

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

Here, \(a = 1\), \(b = 2k\), and \(c = -(2k + 1)\). Plugging in these values:

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

Simplifying this:

\[
\Delta = 4k^2 + 8k + 4
\]

For the curve to touch the x-axis, the discriminant must be zero:

\[
4k^2 + 8k + 4 = 0
\]

This equation can be simplified by dividing everything by 4:

\[
k^2 + 2k + 1 = 0
\]

This can be factored as:

\[
(k + 1)^2 = 0
\]

So, the only solution is:

\[
k = -1
\]

### Conclusion:
The value of \( k \) for which the x-axis is a tangent to the curve \( y = x^2 + (2k)x - (2k + 1) \) is \( k = -1 \).

Do you want more details or have any questions about this solution? Here are five related questions you might consider:

1. How does the discriminant determine the nature of the roots in a quadratic equation?
2. What is the geometric interpretation when the discriminant is positive, zero, or negative?
3. Can you find the point of tangency on the curve for \( k = -1 \)?
4. What happens to the curve when \( k \) is not equal to \(-1\)?
5. How would you derive the equation of the tangent line to the curve at a given point?

**Tip:** Always check the discriminant when solving problems involving the intersection of curves and lines, as it reveals the number and type of solutions (roots).

Find the values of k for which the x-axis is a tangent to the curve y = x^2 + (2k)x - (2k + 1).

Learn how to find values of k for which the x-axis is a tangent to the curve y = x^2 + (2k)x - (2k + 1). This problem involves solving a quadratic equation and understanding the discriminant. Suitable for students in Grades 10-12.

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