We have the equation:

\[
\sqrt{x^2 + 4} = x + 2
\]

### Step 1: Square both sides
To eliminate the square root, square both sides of the equation:

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

This gives:

\[
x^2 + 4 = (x + 2)^2
\]

### Step 2: Expand the right-hand side
Expand \((x + 2)^2\):

\[
(x + 2)^2 = x^2 + 4x + 4
\]

So the equation becomes:

\[
x^2 + 4 = x^2 + 4x + 4
\]

### Step 3: Simplify
Now, subtract \(x^2\) and \(4\) from both sides:

\[
0 = 4x
\]

### Step 4: Solve for \(x\)
From \(0 = 4x\), we find:

\[
x = 0
\]

### Step 5: Check for extraneous solutions
We substitute \(x = 0\) back into the original equation:

\[
\sqrt{0^2 + 4} = 0 + 2
\]
\[
\sqrt{4} = 2
\]

Since both sides are equal, \(x = 0\) is a valid solution.

### Final Solution:
The solution to the equation is \(x = 0\).

Would you like more details or have any questions about the steps? Here are five related questions:

1. How do you determine if a solution is extraneous in radical equations?
2. What happens if you encounter negative values under a square root?
3. Can squaring both sides of an equation introduce new solutions?
4. What is the difference between solving a quadratic equation and a radical equation?
5. Why is it important to check the solution in the original equation after solving?

**Tip:** Always check for extraneous solutions when dealing with square roots in equations, as squaring can introduce false solutions.

Discover how to solve the radical equation √(x^2 + 4) = x + 2 with a comprehensive step-by-step explanation. This problem explores key algebraic concepts such as squaring both sides and checking for extraneous solutions. Ideal for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation