[XCash Prize] Afterburner's Challenges

[Afterburner]

1. Derive the quadratic formula. And by that, I mean "derive" or "prove", not state. ;) 25xcash prize.

2. Prove that if the interior of an ellipse is a mirror, all rays of light emitting from one focus reflect off the inside and pass through the other focus. 50xcash prize.

3. Prove that 0.999~ = 1. 10xcash prize.

That's all for now.

[Posh Jibbons]

You lost me around, "derive."

[Diamond187]

1. Take perfectly generic quadratic:
ax^2+bx+c=0
Complete the square:
a(x+b/2a)^2+c-b^2/4a^2=0
Solve for x:
(x+b/2a)^2=(b^2/4a^2-c)/a
x=sqrt[(b^2/4a^2-c)/a]-b/2a
x=sqrt[(b^2-4ac)/a^2]-b/2a
*magically pulls a 2 out of nowhere*
x=[-b+-sqrt(b^2-4ac)]/2a

I missed a two somehow, and it took me for freakin ever to remeber how to complete a square so I say screw it. AB, just give the cash to whoever finds where I missed the 2.

2. Umm, maybe in writing, not in internet math and certainly not at quarter past 2.

3. This seems to require a limit proof. Iunno if you can do it without a delta-epsilon, but I really don't feel like doing a delta-epsilon right now and I don't see another way to do it other than to glaze over the limit and just say, "trust me, it works".

[John Tennison]

I could be wrong... but I remember seeing someone once prove number 3. And then watching an even smarter math professor prove how they got it wrong. So, I'm gonna go with "It's not actualy possible".

[The First]

I can explain how I'd solve the problem with the ellipse.

To state it with as few words as possible, what you want to prove is that a ray of light will always bounce off of the ellipse walls and travel from one focus to the other, no matter what angle it has. The laws of physics tell us that incoming Angle = Outgoing Angle.

The way I'd solve it is to prove that a line from one focus will always have the same angle as a line coming from the other focus to the same point (or to be exact, 180° - Angle1 = Angle2).

You have your standard equation for an ellipse x²/a² + y²/b² = 1. You only need the derivative of that (the equation for any given tangent on the ellipse) and then you already have everything you need. Now you use trigonometry and prove that the Angles are the same (or, as said, 180° - Angle1 = Angle2).

At least that's how I think it should work, and it makes sense that it should work. If you desperately want the math, I'll do that as well.

[Liokae]

3:

Let x = .9~

x = .9~
10x = 9.9~
10x - x = 9.9~ -.9~
9x = 9
x = 1

[Homer43]

this is obviously not for someone with a 7th grade education :D

[dunce]

ditto

[h2oloobmx]

this is my first post!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!

[h2oloobmx]

well i think it goes like this 1+1=2

[XTTX]

Don't double post h2loobmx and the first post was off topic. Also Liokae I got confused when 9x=9