Topics:
• More on solving radical equations
• What if a “solution” does not check?
• Introduction to complex numbers: imaginary numbers
Examples we worked in class:
• Pairs worked on the second problem in this document: MAT1275RadicalEquations.
One new thing that happened here, which we had not seen before in the examples, was that we ended up having to solve a quadratic equation after squaring both sides, so there were two candidates for solutions to the equation.
When we checked the two proposed solutions in the original equation, only one of them worked. The other one (it was 
• I worked Example 4 from Section 6.7 in the textbook. Here we had to isolate the radical term first. Please note that we did not completely isolate the radical (although we could have done that): it is only necessary to isolate the term which contains the radical, so that when we square both sides, there will be no radical left. For more study, here are notes in which I solve this equation be completely isolating the radical, and I also show you what happens if you try to square both sides without isolating the radical.
In this example, we also ended up solving a quadratic equation along the way, which gave us two candidates for solution to the original equation, and again only one of them worked when we went to check. This is what happened when we went to check the other one:
Look very carefully at what happened here. If we could have taken the negative square root of 81, the next-to-last step would have read
21 – 18 = 3
which is true. Remember that is a symptom of the fact that this proposed solution (y=21) is an “extraneous solution” generated by our solution method, not that we made any error, but rather that the solution method itself sometimes gives extra garbage along with any solutions of the original equation that exist. When checking, we need to look for this and rule such things out.
• Pairs then worked on the Skill Practice problem
Solve
Several points emerged when discussing the student solution that was put on the board: please make sure that you understand them! Here is our complete discussion.
The solution on the board originally gave two values for m, namely &latex m=1$ or 
– so this proposed solution does not check. But is it an “extraneous solution”? NO! Notice that even if we would have been allowed to take the negative square root of 1, this still would not have checked: We would have gotten 
Here are the important points to take away from this (and the previous) examples:
• The method we use to solve equations containing radical expressions will always give any solutions of the equation that exist, but it may also give other numbers as well (sometimes called “extraneous solutions”, but I prefer to just call them “garbage”.) We must check (it’s not just a good idea, it’s the law!) in order to exclude these “garbage solutions” from our final answer. [This is similar to what happens in the case of rational equations.]
• The symptom of a proposed solution being a “garbage solution” which must be excluded, is that when we go to check it in the original equation, it would have worked if the radical sign allowed us to take the negative square root.
• If the proposed solution fails even if we take the negative square root, it means that we made a mistake somewhere and we must go back and correct that mistake! In general, if any proposed solution in any type of equation fails to check, our first thought should be that we may have made an error somewhere and we should carefully check all our work.
• There are two types of equations we have studied in this course for which the solution methods we use can generate “garbage solutions” in addition to any genuine solutions that exist. They are:
rational equations – the symptom is that the proposed solution gives a 0 denominator
radical equations – the symptom is that the proposed solution would work if the radical sign gave the negative square root instead of the positive square root
For any other type of equation in this course, if a proposed solution fails to check, it means that there is an error somewhere.
• In particular, any time we solve an equation by factoring, it is an excellent idea to check the factorization by multiplying it out. Incorrect factoring is a very frequent cause of errors and it is well worth taking the minute or so to check every factorization before proceeding.
• In the cases of solving rational equations or solving radical equations, it is possible that all, some, or none of the proposed solutions will survive to be genuine solutions of the original equation. Anything can happen. It is necessary to check each and every proposed solution in the original equation, and watch for the symptoms of its being a garbage solution which has to be eliminated. If all of the proposed solutions are garbage, the equation has no solutions (and that is our final answer).
New topic: introduction to complex numbers
We have previously said that the square root of a negative real number is not a real number, and we were careful not to say that it is undefined. The reason can now be revealed: it is possible to define square roots of negative real numbers, but those square roots will not be real numbers themselves. We must invent a new type of number in order to do this. They will be called imaginary numbers, even though they are no less real than the real numbers.
Definition:
This is the principal square root of -1. Its name is 
There is another square root of -1, namely 
Because
(Remember, this is the definition of square root.)
To define the square root of any negative real number:
If 
Examples:
Please note the order in which we write those final results. In general, we write numerical factors before the
Homework:
• Review the examples of solving radical equations that we discussed in class. I have left out the details of the two problems you worked in pairs, so that you can re-work them yourselves (and you should do so!) Make sure that you understand all of the points we made in discussing these problems and what happened when we checked them.
• Do the following problems from the Course Outline: section 6.7, p.547 starting with problem 22 (the problems before that have already been done in class or assigned as homework);
• Do section 6.8, p. #11-16 all (this is not in the Course Outline)
• Do the WeBWorK: due by Sunday 11 PM, but do not wait to the last minute!
Remember that you can use the Piazza discussion board to ask questions if you get stuck on any of the WeBWorK or the other homework problems. Don’t forget to include the problem itself in your question, as that will make it easier for you to get a quick response!
