It is important to check all solutions to radical equations because solving these equations can sometimes introduce extraneous solutions—solutions that satisfy the manipulated form of the equation but do not satisfy the original equation. Here's a breakdown of why this happens and the importance of checking:
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Squaring Both Sides: In many cases, solving radical equations involves squaring both sides to eliminate the radical. When you square both sides, you may inadvertently introduce solutions that weren't part of the original equation. For example, if both sides are equal to a negative value, squaring them will yield a positive value, which can lead to false solutions.
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Domain Considerations: Radical expressions have domain restrictions. For instance, the expression \(\sqrt{x}\) is only defined for \(x \geq 0\). When you manipulate equations, you might derive solutions that don't meet these conditions. Checking each solution helps ensure that they are valid in the context of the original equation.
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Real-World Context: In applied problems, extraneous solutions can lead to nonsensical results. For example, if a solution represents a length, it must be a non-negative number. If checking reveals a negative solution, it would indicate an issue with the solution process.
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Prevent Error Propagation: In mathematics, incorrect solutions can propagate through subsequent calculations. By verifying the solutions of radical equations, you can avoid compounding mistakes in later steps or calculations.
In summary, checking all solutions ensures the validity and relevance of the solutions you obtain, both mathematically and in practical applications. This step is crucial for confirming that the answers derived from the manipulation of the equation align with the constraints and properties of the original problem.
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