Short Answer 
An extraneous solution is a result of equation manipulation that does not satisfy the original equation. In the given example, the equation x – 3 = x – 5 leads to potential solutions x = 4 and x = 7, but only x = 7 is valid, making x = 4 the extraneous solution.
Step 1: Understanding Extraneous Solutions
An extraneous solution is a solution that results from the process of solving an equation but does not satisfy the original equation. These solutions can emerge during manipulation and transformations, which may introduce new solutions that are not valid for the original equation. Recognizing which solutions are extraneous is crucial to ensuring accurate results when solving equations.
Step 2: Solving the Equation
To find the extraneous solution, start by solving the equation provided. Here’s the procedure:
- Set the equation: x – 3 = x – 5.
- Square both sides to eliminate the square root: (x Р3)² = (x Р5)².
- Expand and simplify the equation to get: x² Р11x + 28 = 0.
- Factor the quadratic equation: (x – 4)(x – 7) = 0, giving solutions x = 4 and x = 7.
Step 3: Validating the Solutions
Now that we have potential solutions, it is essential to check each one against the original equation:
- For x = 7: Substitute into the original equation to confirm it holds true (4 = 2).
- For x = 4: Substitute and check if the equality holds (1 ≠ 1), indicating that this is not valid.
Since x = 4 does not satisfy the original equation, it is an extraneous solution, while x = 7 is valid.