What is k if   x2 +6x=k has only one  solution? Write the final answer up to two decimals.

To determine the value of \(k\) such that the quadratic equation \(x^2 + 6x = k\) has only **one solution**, we use the concept of the **discriminant**.

A quadratic equation of the form \(ax^2 + bx + c = 0\) has:
- **One solution** (a repeated root) if the discriminant \(b^2 - 4ac = 0\).

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### Step-by-step solution:

1. Rewrite the given equation in the standard form \(ax^2 + bx + c = 0\):  
   \[
   x^2 + 6x - k = 0
   \]
   Here, \(a = 1\), \(b = 6\), and \(c = -k\).

2. The discriminant is:
   \[
   \Delta = b^2 - 4ac = 6^2 - 4(1)(-k)
   \]
   \[
   \Delta = 36 + 4k
   \]

3. For the equation to have only **one solution**:
   \[
   \Delta = 0
   \]
   \[
   36 + 4k = 0
   \]

4. Solve for \(k\):
   \[
   4k = -36
   \]
   \[
   k = -9
   \]

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### Final Answer:
\[
k = -9.00
\]

Would you like to go over anything in more detail, or do you have further questions?  

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Here are 5 related questions for further practice:  
1. What is the discriminant of \(x^2 - 8x + 15 = 0\)?  
2. How many solutions does the equation \(x^2 + 4x + 10 = 0\) have?  
3. How do you solve \(x^2 + 6x = -9\) by factoring?  
4. Can a quadratic equation have no real solutions? If so, when?  
5. How is the vertex of a quadratic function related to its solutions?  

**Tip:** If the discriminant of a quadratic equation is positive, the equation has two distinct real solutions.

What is k if x^2 + 6x = k has only one solution? Write the final answer up to two decimals.

Discover how to find the value of k such that the quadratic equation x^2 + 6x = k has only one solution. This problem explains key algebraic concepts like discriminants and quadratic equations, with a step-by-step solution. Suitable for Grades 8-10.

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