When do you need to check for extraneous roots? (GRE)

For the purposes of the GRE, you only really need to worry about extraneous roots when doing one of two things:

  • squaring a variable
  • getting rid of the absolute value of a variable

If the equation you're given already includes the squared variable (and you don't multiply by the variable yourself), then you don't need to worry about the extraneous root. Both solutions will be correct because the variable was already squared. The trick here is to be very aware of what you choose to do versus what is given to you by the problem.

You only need to worry about the extraneous root in the case of a quadratic equation if you made the equation quadratic by multiplying by a variable. Any time you square a negative number or a variable (which may be negative), you risk losing information by making it positive. That means that, if you square a variable (watch out especially for variables on both sides of the equation, which can easily hide a negative) or a negative number, you should always do a quick plug-in and check.

Here's a post that further covers the topic of extraneous roots: GMAT Math: Algebra Equations with Radicals. You also might find this discussion about GRE problems with negative roots helpful as it also covers relevant information.

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