https://file.aiquickdraw.com/mathbot/file/1726829991951a4oe4has.jpg

Let's break down and solve the problem step by step from the image:

### Given:
The equation of the curve is:
\[ y = (2k - 3)x^2 - kx - (k - 2) \]
where \( k \) is a constant.

The line given is:
\[ y = 3x - 4 \]
which is tangent to the curve. We are tasked with finding the value of \( k \).

### Steps to solve:

1. **Set the two equations equal to each other**:
   Since the line is tangent to the curve, their equations will intersect at exactly one point. Set the equations equal:
   \[
   (2k - 3)x^2 - kx - (k - 2) = 3x - 4
   \]

2. **Simplify to quadratic form**:
   Rearranging the equation:
   \[
   (2k - 3)x^2 - kx - (k - 2) - 3x + 4 = 0
   \]
   Simplify the terms:
   \[
   (2k - 3)x^2 - (k + 3)x - (k - 2 - 4) = 0
   \]
   \[
   (2k - 3)x^2 - (k + 3)x - (k - 6) = 0
   \]
   This is a quadratic equation in the form \( ax^2 + bx + c = 0 \).

3. **Set the discriminant equal to zero**:
   Since the line is tangent, the discriminant of the quadratic equation must be zero for the quadratic to have exactly one solution. The discriminant is given by:
   \[
   \Delta = b^2 - 4ac
   \]
   For the quadratic equation \( (2k - 3)x^2 - (k + 3)x - (k - 6) = 0 \), the coefficients are:
   \[
   a = 2k - 3, \quad b = -(k + 3), \quad c = -(k - 6)
   \]
   Setting the discriminant to zero:
   \[
   (k + 3)^2 - 4(2k - 3)(-(k - 6)) = 0
   \]

4. **Solve the discriminant for \( k \)**:
   Expanding both terms:
   \[
   (k + 3)^2 = k^2 + 6k + 9
   \]
   \[
   -4(2k - 3)(-(k - 6)) = 4(2k - 3)(k - 6)
   \]
   Expand the second term:
   \[
   4(2k - 3)(k - 6) = 4 \left(2k^2 - 12k - 3k + 18\right) = 4(2k^2 - 15k + 18) = 8k^2 - 60k + 72
   \]
   Now substitute into the discriminant equation:
   \[
   k^2 + 6k + 9 - (8k^2 - 60k + 72) = 0
   \]
   Simplify:
   \[
   k^2 + 6k + 9 - 8k^2 + 60k - 72 = 0
   \]
   \[
   -7k^2 + 66k - 63 = 0
   \]
   Divide through by -7:
   \[
   k^2 - 9k + 9 = 0
   \]
   Factorize:
   \[
   (k - 3)^2 = 0
   \]
   So, \( k = 3 \).

Thus, the value of \( k \) is **3**.

Would you like any further details or explanations on any part of this solution?

### Here are 5 related questions to expand understanding:
1. What happens if the line is not tangent but intersects the curve at two points?
2. How do you calculate the discriminant of a quadratic equation in general?
3. What is the geometric significance of setting the discriminant equal to zero?
4. How do different values of \( k \) affect the shape of the parabola given by the equation of the curve?
5. What if the curve was a cubic function? How would you find the value of \( k \)?

### Tip:
Always check the discriminant when dealing with tangency conditions, as it provides key insights into the number of intersections.

The equation of the curve is y = (2k - 3)x^2 - kx - (k - 2), and the line y = 3x - 4 is tangent to the curve. Find the value of k.

This problem involves finding the value of k in the quadratic equation y = (2k - 3)x^2 - kx - (k - 2) where the line y = 3x - 4 is tangent to the curve. The solution applies quadratic equations, tangency conditions, and the discriminant formula.

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