Math Is Fun Forum

Ok, how about this:

Substitution Time!

And put it all together and what do you get:

I understand that, and agree, but these sorts of problems really upset me.

I know they shouldn't, and I know it's irrational, but I kind of feel like problems like these are gimmicky. I wonder how many times in a real-life situation will you come across a problem like that?

Your answer has a certain beauty to it, no doubt. It almost dances as it comes to its conclusion, (as G.H. Hardy said, "The mathematician's patterns, like the painter's or the poet's, must be beautiful"") But I feel like approaching like I did, has more value. I think in real life when you come across an equation very rarely will it be able to be solved with a nifty substitution.

Then again (and I'll contradict myself before someone else does), if I felt that way, why not just graph it and get a "good enough" approximation? Approximations are all you really need in real life.

I guess that's the difference between "Pure Mathematics" and "Applied Mathematics."

Or maybe I'm just jealous that your answer was cooler than mine.

Yeah, I saw that graphically, but I was hoping someone would come along and answer it analytically.

By the way, the other two roots are complex.

3(x²+4x)² - 2(x+2)² =  3(x^4 + 8x³ + 16x²) - 2(x² + 4x + 4) = 3x^4 + 24x³ + 48x² - 2x² - 8x - 8 = 3x^4 + 24x³ + 46x² - 8x - 8 = 57

3x^4 + 24x³ + 46x² - 8x - 65 = 0

Which is a quartic equation that you can read about here.

Well, that doesn't seem so bad. Maybe a bit useless, but not so difficult to learn.

I think that's something I don't understand. The other stuff is all stuff I've seen before, but in a bunch of different classes, Not just one.

Yes, read this wiki on Riemann sums it'll give you an idea of how to go about doing it by hand.

Or you can plug it into a TI-83 or another Graphing program and evaluate it numerically that way.

I’m an American, but might have taken the US equivalent to "Further Maths" What does it entail?

Maybe the reason I haven't seen this before is because I've had minimal experience with vectors. In our class we would simply define a coordinate system and work with x and y. I know what a vector is use, but really didn't use them.

Thanks for clearing that up.

Not to sound like a teacher (I'm most definitely not one) But my rule of thumb is not to let a calc to the work unless I already know how.

In other words, I make a point not to be stupid, but I have no issues with being lazy.

Really? Your method works just fine for me. Where are you getting stuck?

Do you know of any (non-textbooks) books that would be a good place to start with intro to proofs? What's in an intro to proofs class? I've read many proofs (mostly famous ones) and they didn't seem to difficult to follow. Is there something more to it?