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Chapter 7: Problem 7

\(\sqrt{x-2}=3\)

Short Answer

Expert verified
x = 11

Step by step solution

01

Isolate the Square Root

The equation is already isolated as \( \sqrt{x-2}=3 \).
02

Square Both Sides

Square both sides to eliminate the square root. This gives us \( (\sqrt{x-2})^2 = 3^2 \), which simplifies to \( x-2 = 9 \).
03

Solve for x

Add 2 to both sides of the equation to solve for \( x \). This gives us \( x = 9 + 2 \), which simplifies to \( x = 11 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating the Square Root
When solving square root equations, one of the first steps you need to take is to isolate the square root term. In our example, we start with the equation:
\[ \sqrt{x-2} = 3 \]
Notice that the square root is already isolated on one side of the equation. This means there鈥檚 nothing else on the same side as the square root term. Isolating the square root usually involves moving other terms to the opposite side of the equation.
  • Example: If you had \[ \sqrt{x-2} + 5 = 8, \] you would first subtract 5 from both sides to get \[ \sqrt{x-2} = 3. \]
Now you are ready to move on to the next step.
Squaring Both Sides
After you have isolated the square root, the next step is to square both sides of the equation. This is done to eliminate the square root, making it easier to solve for the variable.
  • Example: Starting from our isolated square root:
    \[ \sqrt{x-2} = 3 \]
    Square both sides:
    \[ (\sqrt{x-2})^2 = 3^2 \]
    This simplifes to:
    \[ x-2 = 9 \]
Squaring both sides is essential because it transforms the equation into a simpler form without the square root.
Solving for x
Once you have squared both sides and eliminated the square root, you will need to solve for the variable. In our simplified equation:
\[ x-2 = 9, \]
the next step is to isolate and find the value of \( x \).
  • Add 2 to both sides:
    \[ x-2+2 = 9+2, \]
    Simplifying this gives:
    \[ x = 11 \]
Now, you have found the value of \( x \). It is always good practice to check your solution by substituting it back into the original equation to make sure it holds true:
\[ \sqrt{11-2} = \sqrt{9} = 3. \] The left side equals the right side, verifying your solution is correct.

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