√16+6x-x^2$$ \sqrt{} $$$$ \sqrt{16+6x-x^2} $$

The expression you provided can be interpreted as $ \sqrt{16 + 6x - x^2} $. Here's a breakdown of the expression:

1. **Square Root**: The square root function, denoted by $ \sqrt{} $, applies to the entire expression inside it.
2. **Expression Inside the Square Root**: $ 16 + 6x - x^2 $ is a quadratic expression in terms of $ x $.

### Simplification
The expression inside the square root, $ 16 + 6x - x^2 $, can sometimes be factored or simplified. Let's try to factor it.

1. Rewrite $ -x^2 + 6x + 16 $ as:
   \[
   - (x^2 - 6x - 16)
   \]
2. Check if $ x^2 - 6x - 16 $ can be factored. Unfortunately, $ x^2 - 6x - 16 $ does not factor neatly, so we'll work with it as it is.

Thus, the simplified expression remains $ \sqrt{16 + 6x - x^2} $.

### Domain Consideration
For the expression $ \sqrt{16 + 6x - x^2} $ to be defined, the expression inside the square root must be non-negative:
\[
16 + 6x - x^2 \geq 0
\]
This inequality will define the domain of the function.

### Factoring for Domain
We complete the square to find the roots and determine the interval:
1. Rewrite the quadratic as:
   \[
   -\left(x^2 - 6x + 16\right) = 0
   \]
   Add and subtract $\left(\frac{6}{2}\right)^2 = 9$:
   \[
   -\left((x - 3)^2 - 25\right) = 0
   \]
2. So, $ 16 + 6x - x^2 = 25 - (x-3)^2 $.

This shows that the square root will be real when $ (x-3)^2 \leq 25 $, meaning $ -2 \leq x \leq 8 $.

### Conclusion
The expression $ \sqrt{16 + 6x - x^2} $ is defined for $ x $ in the interval $[-2, 8]$. 

Would you like to explore the domain further, or do you have any specific questions?

### Questions for Further Exploration:
1. How do you factor quadratic expressions?
2. What does it mean to complete the square?
3. How do you find the domain of a square root function?
4. Can this expression be plotted on a graph?
5. What happens to the expression outside of the domain?

**Tip:** When simplifying square roots with quadratic expressions, always check if the quadratic can be factored or completed to ensure the correct domain and simplify your work.

Explore how to simplify and determine the domain of square roots involving quadratic expressions. Learn about factoring quadratic equations and completing the square to define the domain of the square root function. Suitable for students in Grades 10-12.

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