Extraneous Or No Solutions--Example 5

Topic

Equations

Description

This example presents an equation involving both radicals and rational expressions, which are particularly prone to producing extraneous solutions. The equation might take a form like the one shown above. Solving such equations typically involves a multi-step process. First, one might need to rationalize both sides of the equation. This often involves squaring both sides of the equation, sometimes multiple times, to eliminate all radicals. These operations can inadvertently introduce extraneous solutions. After simplification, the resulting equation is likely to be of a higher degree, possibly quartic or higher, requiring advanced solving techniques. The critical step in this example would be the meticulous verification of solutions. Each solution obtained must be carefully checked by substituting it back into the original equation. This process often reveals that many algebraically correct solutions fail to satisfy the original equation due to domain restrictions or the introduction of extraneous solutions during the solving process.

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