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Check equation form and domain, find candidates, and reject those failing the original equation.

Meaning

Equations with denominators or radicals, and higher-degree equations, require domain checks and suitable transformations. If a transformation is not equivalent, test the candidates in the original equation.

Example

Squaring gives with candidates 2 and -1. At -1, the original sides are 1 and -1, so it is rejected; the solution is 2.The square-root side is nonnegative, so the original equation also requires x≥0. Squaring loses the distinction between positive and negative values, introducing a candidate such as −1; this is why the original equation must be checked.Squaring or clearing denominators can introduce extraneous candidates. Values making a denominator zero must also be excluded.

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Scope and references

Topics

Basic cubic and quartic forms, simultaneous quadratic equations, rational and radical equations

Learning goal

Check equation form and domain, find candidates, and reject those failing the original equation.

Helpful background

Check factors, denominators, and radical restrictions for the given form.

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These ordinary choice questions provide limited practice in this concept. They do not establish independent solution, full-concept performance, or mastery.

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