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Is there a systematic way of preventing false roots when squaring a root equation? The testing of the roots is quite tedious in some problems.

My first thought was absolute values in some form

user262493
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  • It is often useful to square, possibly introduce extraneous roots, and get rid of them later. That can be more efficient than working directly with absolute values. – André Nicolas Aug 23 '15 at 18:59
  • The term you're looking for is "extraneous" and there's a couple tricks to identifying them. For example, you could plug in the solution into the square root terms and check if you get a negative number under the square root. Then you immediately know that's a source of an extraneous solution. – user217285 Aug 23 '15 at 19:00
  • Do you have an example? I'm having a hard time thinking of a problem where it is harder to test roots then it is to find them in the first place – pancini Aug 23 '15 at 19:01

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Let $A$ and $B$ denote mathematical expressions which evaluate to real numbers. There are two cases:

If you have an equation of the form $\sqrt{A} = B$, it is equivalent to $A = B^2, B \geq 0$.

If you have an equation of the form $\sqrt{A} = \sqrt{B}$, it is equivalent to $A = B \geq 0$.

So basically you have to transform one equation into an equation plus an inequality to have an equivalence transform.

Alex1357
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