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To find the values of \( k \) for which the quadratic equation \( 7x^2 - 9x + k = 0 \) has exactly one real solution, we need to look at the discriminant of the quadratic equation. For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant (\( \Delta \)) is given by:

\[
\Delta = b^2 - 4ac
\]

The equation has:
- **One real solution** if and only if \( \Delta = 0 \).

For the given equation \( 7x^2 - 9x + k = 0 \):
- \( a = 7 \)
- \( b = -9 \)
- \( c = k \)

Substitute these into the discriminant formula:

\[
\Delta = (-9)^2 - 4 \times 7 \times k
\]

\[
\Delta = 81 - 28k
\]

Set the discriminant equal to zero to find the condition for one real solution:

\[
81 - 28k = 0
\]

Solve for \( k \):

\[
81 = 28k
\]

\[
k = \frac{81}{28}
\]

So, the value of \( k \) for which the quadratic equation has exactly one real solution is:

\[
k = \frac{81}{28}
\]

Would you like more details on how this value was derived or have any further questions?

Here are 5 related questions you might find interesting:
1. What happens to the quadratic equation if \( k \) is greater than \( \frac{81}{28} \)?
2. What are the values of \( k \) for which the equation has two real solutions?
3. How can you express the solution in decimal form instead of fraction form?
4. Can you find the value of \( x \) when \( k = \frac{81}{28} \)?
5. How does changing the coefficient of \( x^2 \) in the equation affect the discriminant?

**Tip:** The discriminant is a powerful tool in determining the nature of solutions of quadratic equations. Remember, \( \Delta > 0 \) means two distinct real solutions, \( \Delta = 0 \) means one real solution, and \( \Delta < 0 \) means no real solutions.

Find all values of k for which the quadratic equation 7x^2 - 9x + k = 0 has one real solution.

Solve the quadratic equation 7x^2 - 9x + k = 0 to find the value of k for which the equation has exactly one real solution. This problem involves using the discriminant formula and is suitable for students in Grades 9-11.

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