Article: Recall that after isolating the radical on one side of the equation, you then squared both sides to remove the radical sign. This is a necessary step to solving the problem. However, the squaring operation is what creates the extraneous solutions. Remember some basic mathematics, that both a negative and a positive number, when squared, will give the same result. For example, (3)2{\displaystyle (-3)^{2}} and 32{\displaystyle 3^{2}} both give the answer of 9{\displaystyle 9}. However, both the negative and positive numbers might not be solutions to whatever problem you are solving. The one that does not work is called the extraneous solution. After you have found the solutions to your problem, you may have found one, two or more different possible values for the variable. You need to check each of these in the original problem to see which work. Remember that the original problem here was x1+4=x3{\displaystyle {\sqrt {x-1}}+4=x-3}. First check the solution x=5{\displaystyle x=5}: x1+4=x3{\displaystyle {\sqrt {x-1}}+4=x-3} 51+4=53{\displaystyle {\sqrt {5-1}}+4=5-3} . (substitute 5 for x) 4+4=53{\displaystyle {\sqrt {4}}+4=5-3} 2+4=53{\displaystyle 2+4=5-3} 6=2{\displaystyle 6=2}. Because your result is an incorrect statement, the original solution of x=5{\displaystyle x=5} must be an extraneous solution that was caused by the squaring process. Check the second solution x=10{\displaystyle x=10}: x1+4=x3{\displaystyle {\sqrt {x-1}}+4=x-3} 101+4=103{\displaystyle {\sqrt {10-1}}+4=10-3} 9+4=103{\displaystyle {\sqrt {9}}+4=10-3} 3+4=103{\displaystyle 3+4=10-3} 7=7{\displaystyle 7=7} In this case, you get a true statement. This shows that the solution x=10{\displaystyle x=10} is a true solution to the original problem. The extraneous solution is incorrect and can be discarded. Whatever remains is the answer to your problem. In this case, you would report that x=10{\displaystyle x=10}.

What is a summary?
Recognize the potential for an extraneous solution. Test each of your solutions in the original problem. Discard the extraneous solution and report your result.