The problem asks for the two values of $k$ such that two different quadratic functions of the form $y = 9x^2 - kx + 4$ each touch the x-axis only once.

For a quadratic equation to touch the x-axis only once, it must have a double root. This means the discriminant of the quadratic must be zero. The general form of a quadratic equation is:

$y = ax^2 + bx + c$

Here, $a = 9$, $b = -k$, and $c = 4$. The discriminant $\Delta$ is given by:

$\Delta = b^2 - 4ac$

Substituting $a = 9$, $b = -k$, and $c = 4$ into the discriminant formula:

$\Delta = (-k)^2 - 4(9)(4)$ $\Delta = k^2 - 144$

For the quadratic to touch the x-axis only once, we need $\Delta = 0$:

$k^2 - 144 = 0$

Solve for $k$:

$k^2 = 144$ $k = \pm 12$

Thus, the two values of $k$ are $k = 12$ and $k = -12$.

Would you like more details, or do you have any questions?

Here are 5 related questions:

1. What is the geometric interpretation of a double root in a quadratic function?
2. How does the discriminant help determine the nature of the roots of a quadratic?
3. What would happen if $k^2 - 144 > 0$? What would this imply about the roots?
4. How would the quadratic graph change if the discriminant were positive or negative?
5. Can you solve for $k$ if the quadratic had distinct roots instead of a double root?

Tip: When solving quadratic equations, the discriminant is a quick way to check the nature of the roots (real, imaginary, or repeated).