A False Answer
Part of the Radicals & Rational Exponents section of Coddy's Algebra 2 journey. Lesson 27 of 43.
Squaring makes two different numbers look alike: -3 and 3 are not equal, but their squares are both 9. So squaring can turn a false equation into a true one, and the squared equation may have answers the original does not.
Take √(x + 6) = x. Squaring gives x + 6 = x^2, so x^2 - x - 6 = 0, which factors as (x - 3)(x + 2) = 0. The candidates are 3 and -2. Check each in the original. For 3: √(9) is 3, and the right side is 3. True. For -2: √(4) is 2, but the right side is -2. False. So -2 is thrown out, and the only answer is 3.
A candidate that the working produces but the original equation rejects is a false answer (textbooks say extraneous). It is not a slip in your algebra: the squaring brought it in. That is why every candidate is checked in the original equation, never in the squared one, where both pass.
Challenge
Solve √(x + 7) = x - 5. Squaring gives two candidates: check both in the original equation and keep only the one that works.
Try it yourself
Solve for x
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Radicals & Rational Exponents
1Simplifying Radicals
Roots and PowersPerfect Square FactorsThe Largest SquareLetters Under the RootCube RootsRecap: Simplifying Radicals7Powers and Roots Equations
Odd PowersEven PowersFractional PowersTwo Thirds PowerRecap: Power Equations3Multiplying Radicals
Root Times RootDistributing a RootTwo BracketsConjugate PairsRecap: Multiplying Radicals6Radical Equations
Square Both SidesIsolate the Root FirstA False AnswerCube Root EquationsRecap: Radical Equations