Perform the indicated operation and simplify. Assume that the variable represents a nonnegative real number.
left parenthesis 2 minus StartRoot x plus 6 EndRoot right parenthesis squared

We are given the expression:

\[
(2 - \sqrt{x + 6})^2
\]

We need to expand and simplify it. To do this, we can apply the binomial expansion formula:

\[
(a - b)^2 = a^2 - 2ab + b^2
\]

In our case, \(a = 2\) and \(b = \sqrt{x + 6}\). Substituting these into the formula, we get:

\[
(2 - \sqrt{x + 6})^2 = 2^2 - 2(2)(\sqrt{x + 6}) + (\sqrt{x + 6})^2
\]

Now, let's simplify each part:

1. \(2^2 = 4\)
2. \(-2(2)(\sqrt{x + 6}) = -4\sqrt{x + 6}\)
3. \((\sqrt{x + 6})^2 = x + 6\)

Putting everything together:

\[
(2 - \sqrt{x + 6})^2 = 4 - 4\sqrt{x + 6} + (x + 6)
\]

Finally, simplify the constants:

\[
4 + x + 6 - 4\sqrt{x + 6} = x + 10 - 4\sqrt{x + 6}
\]

Thus, the simplified expression is:

\[
x + 10 - 4\sqrt{x + 6}
\]

Would you like any additional details or further clarification?

Here are 5 related questions:
1. How do you apply the binomial expansion to expressions with square roots?
2. What is the significance of assuming a nonnegative real number for \(x\)?
3. How would the solution change if the sign inside the parentheses were reversed?
4. Can you apply this method to expressions like \((a - b)^3\)?
5. How do we simplify higher-degree polynomials with square roots?

**Tip:** Always check for opportunities to combine like terms when simplifying expressions to ensure you have the simplest form.

Perform the indicated operation and simplify. Assume that the variable represents a nonnegative real number.
(2 - √(x + 6))^2

Learn how to expand and simplify the expression (2 - √(x + 6))^2 step-by-step using the binomial expansion formula. This problem demonstrates the application of algebraic methods, suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation