Step 1: Understanding the Concept:
This is a radical equation. To solve for a variable locked inside a square root, we must isolate the radical expression and then perform the inverse operation of a square root, which is squaring.
Squaring both sides of an equation preserves the equality, though it can sometimes introduce extraneous solutions (which must be verified).
Step 2: Key Formula or Approach:
Standard Algebraic Property: \( (\sqrt{a})^2 = a \).
Property of Equality: If \( a = b \), then \( a^2 = b^2 \).
Step 3: Detailed Explanation:
The provided equation is:
\[ \sqrt{x + 9} = 5 \]
To eliminate the radical sign, we square both sides of the equation:
\[ (\sqrt{x + 9})^2 = (5)^2 \]
As per the rules of exponents, the square cancels the square root on the left side:
\[ x + 9 = 25 \]
Now, we perform a simple subtraction to isolate the variable \( x \):
\[ x = 25 - 9 \]
\[ x = 16 \]
Verification:
Let's plug \( x = 16 \) back into the original radical expression:
\[ \sqrt{16 + 9} = \sqrt{25} = 5 \]
The result matches the right-hand side (RHS) of the original equation, confirming the value is correct.
Step 4: Final Answer:
The value of \( x \) is 16.