a ninth grade algebra problem

by MaxEntropy_Man Sat Sep 22, 2012 5:45 pm

simplify:

sqrt[2b+2 sqrt[b^2-4]]/(sqrt[b^2-4]+b+2)

an article of interest to parents of high schoolers with an interest in mathematics (the calculus trap):

http://www.artofproblemsolving.com/Resources/Papers/CalculusTrap.pdf

by aanjaneya Sat Sep 22, 2012 6:43 pm

1 + \sqrt{\frac{b-2}{b+2} }

by Guest Sun Sep 23, 2012 6:15 am

The expression simplifies to 1/sqrt(b+2). Even though this expression is defined for b>-2, the domain should be restricted to b>=2 in order to be consistent with the domain of the original expression.

by MaxEntropy_Man Sun Sep 23, 2012 8:27 am

blabberwock wrote:The expression simplifies to 1/sqrt(b+2). Even though this expression is defined for b>-2, the domain should be restricted to b>=2 in order to be consistent with the domain of the original expression.

correct.

by aanjaneya Sun Sep 23, 2012 8:37 am

You are right.

I had missed the outer most \sqrt in the numerator of the starting expression. (Try doing so, you will indeed get what I have above.)

About the domain restriction qualifier, you are assuming that the original expression needs to live on the Real Line.
If it is allowed to live on the Extended Real Line, or further live on the Extended Complex Plane, then we will be okay without any such caveats.
Extended refers to allowing +/- Infinity, in addition to Real values.

But for a 9th class audience, you are again right in observing it with necessary rigor.

by MaxEntropy_Man Sun Sep 23, 2012 8:41 am

i had failed to mention that the original problem explicitly specifies the domain to be restricted to R although the kids in the class have been taught about solutions in C.

by Guest Sun Sep 23, 2012 9:26 am

Max,

Are you using the AoPS textbooks, online classes and videos for your kids?

We use their material for our children and find them quite useful.

Thanks,
BW

by MaxEntropy_Man Sun Sep 23, 2012 9:33 am

blabberwock wrote:Max,

Are you using the AoPS textbooks, online classes and videos for your kids?

We use their material for our children and find them quite useful.

Thanks,
BW

the older one's math teacher at school, a very serious and thoughtful woman in charge of the mathematics education of a small group of highly motivated kids with an aptitude for challenging math introduced us to the website. she uses these resources in her class. we are fortunate to have her as our teacher.

she has changed my views about high school math education a bit. instead of galloping ahead with topics, she takes the time to explore topics in depth and works on complex and challenging problems in topics they've already mastered before moving on to the next topic.

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