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Chapter 3: Problem 35

Solve. $$\sqrt{7-x}=2$$

Short Answer

Expert verified
x = 3

Step by step solution

01

Isolate the Square Root

The equation is already isolated with the square root on the left-hand side: \( \sqrt{7-x} = 2 \)
02

Square Both Sides

To eliminate the square root, square both sides of the equation: \[ (\sqrt{7-x})^2 = 2^2 \] This simplifies to: \[ 7 - x = 4 \]
03

Solve for x

To find the value of \( x \), subtract 4 from both sides of the equation: \[ 7 - 4 = x \] This simplifies to: \[ x = 3 \]
04

Verify the Solution

To ensure the solution is correct, substitute \( x = 3 \) back into the original equation: \[ \sqrt{7-3} = 2 \]This simplifies to: \[ \sqrt{4} = 2 \], which is true.

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Key Concepts

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

Isolating Square Roots
When solving an equation involving a square root, your first step is usually to isolate the square root on one side of the equation. This ensures you can easily eliminate the square root in the next step. For example, if you start with an equation like \( \sqrt{7-x} = 2 \), the square root is already isolated on the left-hand side. When the square root is on its own, move on to the next step.
Squaring Both Sides
Once the square root is isolated, you need to square both sides of the equation to eliminate it. Remember, whatever operation you perform on one side of the equation, you must also do to the other. In our example, we started with \( \sqrt{7-x} = 2 \). Next, we square both sides:
\[ (\sqrt{7-x})^2 = 2^2 \]
This simplifies to:
\[ 7 - x = 4 \]
This step is crucial because it removes the square root, allowing us to work with a simpler equation to solve.
Solution Verification
After solving the equation, it鈥檚 important to verify the solution. This step ensures that no mistakes were made during the solving process and that the solution is valid. In our example, we found that \( x = 3 \). To verify this, substitute \( x \) back into the original equation:
  • Start with the original equation: \( \sqrt{7-3} = 2 \)
  • Simplify inside the square root: \( \sqrt{4} = 2 \)
  • Confirm the equality: \( 2 = 2 \)
If the original equation holds true with your solution, then you have verified it correctly.

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