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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).

Two different quadratic functions of the form y = 9x^2 - kx + 4 each touch the x-axis only once. Find the two values of k.

Learn how to find the two values of k for the quadratic function y = 9x^2 - kx + 4 such that it touches the x-axis only once. This problem explores discriminants and quadratic equations, ideal for students in Grades 9-11.

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