The frown is due to the origin of infinity. Computer application in mathematics has blurred the lines of what is and isn't acceptable when working with infinity.

Tumbling down this rabbit hole will lead us to discussing the existence of God himself.

chocobo7779 what level are you sitting for?

If you're looking to hone your skills, I suggest the following:
1. Be a little more proactive. Pick a topic, and work on it first. The forum should be your last resort, not the first place you reach for help.
2. Don't be afraid to make mistakes - it's our best tool for learning.

Last but not least, a reminder to all who ask for help in this thread - some of the methods here aren't exactly what your teacher/lecturer prefers. If you're sitting for an internal examination (college course etc.), do check if your lecturer accepts the given solutions (some lecturers can be fussy about these kind of things, so be wary).

Expansion of trinomials was part of your STPM syllabus, so it should be manageable, no? Just use Pascal's triangle!

Just multiply one by one, don't group them together. You should get .

Tangent = intersects at only ONE point.
Please recall your coordinate geometry.

Trinomial expansion is covered just before Binomial expansion in the STPM syllabus.

It hasn't been removed. It follows the same basic principles as regular expansion, just more elements. You'll be fine as long as you're careful with the multiplications.

I'll give you a simple example: Addition!
1+1=2, but what is 1+2?
1+2 is in fact 1+1+1!

Mathematics operates on extensibility, i.e. the ability to extend existing results to form newer ones. Just like the addition example above, your expansion is extensible, i.e. you proceed in the same fashion as you would regular binomial expansion, just extended to more variables.

Listen to Critical_Fallacy. Being proactive is very important, be it in studies, or at the workplace.

On a separate note, Wolfram shows you step-by-step computational work (you can see the button in Critical's picture). If you have problems with understanding the expansion, play around with the variables and see what the results are.

Agreed.

My bad. It used to be free, looks like they've slapped a subscription cost on it now.

No. Use the discriminant and the fact that y=mx+c is the tangent. This information was given in the question for a reason.

You're right, it's not covered in the SPM syllabus. As I've mentioned earlier, using methods which aren't specifically mentioned in the syllabus isn't recommended for exams (it's fair game elsewhere). I know for a fact that many of the examiners these days are new, and tend to be very strict when it comes to following the answer scheme.

I've seen too many students pressing on shortcuts and pretty answers that they end up making mistakes and/or not understanding what they're actually doing. When you become comfortable with your workings, that elegance you seek will come naturally.

I've not tried computing it, but some of the terms should cancel out.

Looks like you're right.

We'll go back to using the sum and product of roots then.

Based on the quadratic equation you gave in Post #1074,
Sum of roots = 2alpha = -(2mc-2mb-2a)
Product of roots = alpha^2 = a^2 +b^2 -2bc-c^2 -r^2

Thus alpha = a+mb-mc and (a+mb+mc)^2 = a^2 +b^2 -2bc-c^2 -r^2. Does this reduce to the answer you seek?

+1 for noticing that. I almost always reply to this thread in between games of DotA, so you'll have to excuse my tardiness.

If you can't get the final answer, look back to the quadratic equation to make sure there are no errors there that may have been carried on.

Here's a good one, courtest of /k/.


maximR you may find this one interesting.

It is because of the periodic nature of trigonometric functions, as kingkingyyk has demonstrated. Do you remember your acute/obtuse angles?

Well done!

Never really gave much thought to it. I do however have something to ask you:

Suppose we have a manifold M immersed in space-time. What would the physical interpretation of the manifold's tangent space be?

I think sometimes you forget that I'm doing my PhD in maths.

We teach general topology as an undergraduate course, and point set topology in the postgraduate classes.

Not really. I rarely deal with numerical methods as I'm an information geometrist. I generally work with the differential-geometric properties of statistical manifolds.