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Topic: Solve For X. (Read 679 times) |
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rloginunix
Uberpuzzler
Posts: 1029
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Solve For X.
« on: Oct 25th, 2014, 11:09am » |
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Solve For X:
( (2 +3))x + ((2 - 3))x = 2x
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dudiobugtron
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Posts: 735
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Re: Solve For X.
« Reply #1 on: Oct 25th, 2014, 7:27pm » |
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lol, I think the answer is x = 2
This is a cool puzzle.
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| « Last Edit: Oct 25th, 2014, 7:28pm by dudiobugtron » |
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rloginunix
Uberpuzzler
Posts: 1029
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Re: Solve For X.
« Reply #2 on: Oct 26th, 2014, 8:08am » |
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The answer is correct but what about the proof that there are no other solutions? That's the interesting part.
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dudiobugtron
Uberpuzzler
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Re: Solve For X.
« Reply #3 on: Oct 26th, 2014, 2:02pm » |
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Graphically, we can see that ((2 +3))x + ((2 - 3))x - 2x is always decreasing. You can also differentiate it and graph that to show that the gradient is always negative.
I imagine that's not the interesting way of solving it you were talking about though!
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| « Last Edit: Oct 26th, 2014, 2:02pm by dudiobugtron » |
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rloginunix
Uberpuzzler
Posts: 1029
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Re: Solve For X.
« Reply #4 on: Oct 27th, 2014, 7:27am » |
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The approach I have in mind is rather simple - it has a trig to it.
Start by dividing both sides of the equation by 2x.
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towr
wu::riddles Moderator
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Re: Solve For X.
« Reply #5 on: Oct 27th, 2014, 9:45am » |
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And then notice one side is strictly monotonically increasing and the other side constant and therefore only cross in one point?
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Wikipedia, Google, Mathworld, Integer sequence DB
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rloginunix
Uberpuzzler
Posts: 1029
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Re: Solve For X.
« Reply #6 on: Oct 27th, 2014, 11:37am » |
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That is one way of doing it. The other, a bit more explicit, is to recall that you already know what x is. Keeping the trigonometric clue in mind we have: something squared plus something squared equals one ...
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SWF
Uberpuzzler
Posts: 879
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Re: Solve For X.
« Reply #7 on: Oct 27th, 2014, 7:59pm » |
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I see what rloginunix is looking for: cos(pi/12)^x+sin(pii/12)^x=1, and left hand side is a decreasing function of x, so 1 solution.
Here is another way: let y=x/2:
(2+sqrt(3))^y + (2-sqrt(3))^y = 4^y
Divide by 2+sqrt(3) and rearrange to get:
1+zy = (1 + z)y
where z is 7-4*sqrt(3), and is between 0 and 1.
Left hand side decreases with y and right hand side increases with y, so 1 solution (y=1).
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rloginunix
Uberpuzzler
Posts: 1029
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Re: Solve For X.
« Reply #8 on: Oct 28th, 2014, 8:36am » |
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You got it, SWF.
The argument to Sin and Cos is either  /12, as you have it, or, if you switch the substitutions around, 5/12 but the numbers come out the same.
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dudiobugtron
Uberpuzzler
Posts: 735
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Re: Solve For X.
« Reply #9 on: Oct 28th, 2014, 5:19pm » |
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There's a more general case to this, although it uses towr's/SWF's idea rather than a trig identity!
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For any numbers a, b and c with 0 < a,b < c
a^x + b^x = c^x
has at most one solution for x.
proof: divide both sides by c^x
Then you have (a/c)^x + (b/c)^x = 1
a/c and b/c are both less than 1, so the left hand side is strictly monotonically decreasing. Therefore, there's at most one solution.
(It's pretty easy to show using the intermediate value theorem that there will be at least one solution; that's an exercise left to the reader though!)
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