The expression you provided is \( \sqrt{16 + 6x - x^2} \). Let's analyze and simplify it step by step.

### Step 1: Rewrite the Expression Inside the Square Root
We start by rearranging the terms inside the square root to match a standard quadratic form:
\[
\sqrt{-x^2 + 6x + 16}
\]
This can be rewritten as:
\[
\sqrt{-(x^2 - 6x - 16)}
\]

### Step 2: Completing the Square
To better understand and potentially simplify the expression, we can complete the square:

1. Consider the quadratic part \( x^2 - 6x \).
2. The expression \( x^2 - 6x \) can be rewritten by completing the square:
   \[
   x^2 - 6x = \left(x - 3\right)^2 - 9
   \]
3. Substitute this back into the expression:
   \[
   \sqrt{-\left[\left(x - 3\right)^2 - 9 + 16\right]}
   \]
4. Simplify further:
   \[
   \sqrt{-\left[\left(x - 3\right)^2 + 7\right]}
   \]

### Step 3: Analyzing the Expression
Now the expression under the square root is:
\[
\sqrt{-\left[\left(x - 3\right)^2 + 7\right]}
\]
This is where we should stop to analyze the expression. The term inside the square root, \( -\left[\left(x - 3\right)^2 + 7\right] \), is always negative for all real \( x \) because \( \left(x - 3\right)^2 \) is always non-negative, and adding 7 makes it strictly positive. The negative sign in front of the entire expression makes it strictly negative.

### Conclusion
Since the square root of a negative number is not a real number, the expression \( \sqrt{16 + 6x - x^2} \) does not yield real values for any real number \( x \). In the real number system, this expression is undefined.

Would you like to explore this further in the context of complex numbers, or do you have other questions?

### Questions for Further Exploration:
1. What happens when you take the square root of a negative number?
2. How do complex numbers come into play with square roots of negative numbers?
3. Can this expression be graphed on the real number plane?
4. What is the significance of completing the square in solving quadratic equations?
5. How do you determine the domain of a function involving a square root?

**Tip:** When dealing with square roots, always check if the expression under the square root is non-negative to determine if the function is real-valued.

Explore how to simplify the expression sqrt(16 + 6x - x^2) step-by-step with detailed explanations and solutions. Understand the concepts of square roots, quadratic equations, and completing the square. Suitable for advanced high school students and anyone interested in algebraic manipulation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation