We are given the line $y = 2x - 3$ and the curve $y = x^2 + kx + 6$, and we are asked to find the possible values of $k$ such that the line touches the curve. This means the line is tangent to the curve, implying they intersect at exactly one point.

Step 1: Set the equations equal to find the point(s) of intersection

We equate the expressions for $y$ from the line and the curve:

$2x - 3 = x^2 + kx + 6$

Step 2: Rearrange the equation to form a quadratic

Move all terms to one side:

$x^2 + (k - 2)x + 9 = 0$

This is a quadratic equation in $x$, and for the line to be tangent to the curve, this quadratic must have exactly one solution. For a quadratic to have one solution, its discriminant must be zero.

Step 3: Apply the discriminant condition

The discriminant $\Delta$ of a quadratic equation $ax^2 + bx + c = 0$ is given by:

$\Delta = b^2 - 4ac$

For the quadratic $x^2 + (k - 2)x + 9 = 0$, we have:

• $a = 1$,
• $b = k - 2$,
• $c = 9$.

Substituting into the discriminant formula:

$\Delta = (k - 2)^2 - 4(1)(9) = (k - 2)^2 - 36$

Step 4: Set the discriminant to zero

For the quadratic to have exactly one solution, the discriminant must be zero:

$(k - 2)^2 - 36 = 0$

$(k - 2)^2 = 36$

Step 5: Solve for $k$

Take the square root of both sides:

$k - 2 = \pm 6$

This gives two possible solutions for $k$:

$k - 2 = 6 \quad \text{or} \quad k - 2 = -6$

Solving for $k$:

$k = 8 \quad \text{or} \quad k = -4$

Final Answer:

The possible values of $k$ are $\boxed{8}$ and $\boxed{-4}$.

Would you like more details on any step, or have further questions?

Here are 5 related questions to explore:

1. What is the geometric significance of the discriminant being zero?
2. How would the solution change if the line and curve intersected at two points?
3. Can the tangent condition be used for curves other than parabolas?
4. How do you find the point of tangency once you know $k$?
5. What happens if the discriminant is negative in such problems?

Tip: The discriminant of a quadratic equation helps determine the number and nature of solutions—real and distinct, real and equal, or complex!