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Mathematics LibreTexts

6.3.1: Resources and Key Concepts

  • Page ID
    197603
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    Key Concepts

    Definitions

    • Radical Equation: An equation in which unknowns (variables) are part of a radicand.
    • Extraneous Solution (Spurious Solution): A root of a transformed (manipulated) equation that is not a root of the original equation because it was excluded from the domain of the original equation or introduced by operations like squaring both sides.
    • Contradiction: A statement that is both true and false simultaneously (e.g., arriving at \(2 \neq 0\) after assuming a solution exists and checking it).

    Common Mistakes

    • Forgetting to Check for Extraneous Solutions: When raising both sides of an equation to an even power (e.g., squaring), it is crucial to check all candidate solutions in the original equation, as this process can introduce extraneous solutions.
    • Incorrectly Isolating the Radical: Failing to isolate the radical term on one side of the equation before raising both sides to a power. This leads to more complicated expressions that still contain radicals.
    • Errors in Raising Both Sides to a Power: Not raising the entire side to the power, especially if it's a binomial (e.g., if \(\sqrt{A} = B+C\), then \(A = (B+C)^2\), not \(B^2+C^2\)).
    • Errors with Rational Exponents: Incorrectly applying powers to "undo" rational exponents (e.g., for \(x^{m/n}=k\), raise both sides to \(n/m\)).
    • Not Recognizing Equations Quadratic-in-Form: Failing to see that a radical or rational exponent equation can be solved using u-substitution if it's quadratic-in-form.

    This page titled 6.3.1: Resources and Key Concepts is shared under a not declared license and was authored, remixed, and/or curated by Roy Simpson.

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