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).