SOLUTION: What are extraneous solutions of an equation? Why do they sometimes occur in the process of solving rational or radical equations? Provide examples to support your answer.

Question 372828: What are extraneous solutions of an equation? Why do they sometimes occur in the process of solving rational or radical equations? Provide examples to support your answer.
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
Extraneous solutions occur because we perform extraneous operations. For example , solve . Some would mulitply both sides by x-4 (which is an extraneous operation). This would give 2(x-4)+5 = x+1,
2x - 8 + 5 = x+1, which would give x = 4. BUT we know that x = 4 is not in the domain of the equation, hence cannot be a solution.
Another equation is . What we normally do is square both sides after transposing 1 to the other side (which is essentially an extraneous operation). It would give the equation , which after simplification gives, and the candidates -1/4 and 1. Unfortunately, 1 does not satisfy the original equation (-1/4 does!) The possible root x = 1 was "introduced" in the part where we squared both sides of the equation.
Hope you understood that, student...
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