Advanced Math / Radical And Rational Equations
Extraneous solutions and domain checks
Radical and rational equations often produce candidates that must be checked. Squaring can introduce extraneous solutions, and denominators cannot be zero.
Least you need to know
- Check candidates in the original equation.
- A denominator of 0 is never allowed.
- A square root output is nonnegative.
Key notation
- \sqrt{x} — The principal nonnegative square root
Worked example
Solving $\sqrt{x}=x-2$ by squaring gives candidates. Each must be substituted back into the original equation because squaring can create invalid values.
Common mistakes
- Canceled factors can hide excluded values.
- A squared equation can have more solutions than the original.
How to recognize it
- The equation contains a square root or a variable in a denominator.
- A candidate makes a denominator zero or a square root side negative.
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