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May 14th, 2012, 11:10 PM   #1
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find x

?( (1-?(1-x^2))/2 ) + ?( (1+?(1-x^2))/2 ) = ?(1+x)
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May 15th, 2012, 12:09 AM   #2
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Re: find x

We are given:



We see we require in order for the left side to represent real values.

Multiply through by :



Square both sides since they are non-negative:









Case 1:





Hence, we find:



Case 2:





No solution for this case, thus we are left with:

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May 15th, 2012, 12:35 AM   #3
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Re: find x

Thanks a lot!
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May 15th, 2012, 06:13 AM   #4
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Re: find x

A sorta "short cut to typing!" I'd use here:
since x^2 - 1 = (x+1)(x-1): let a = x+1 and b = x-1; so, after multiplication by SQRT(2), you have:
SQRT[1 - SQRT(ab)] + SQRT[1 + SQRT(ab)] = SQRT(2a)

Easier to do the squaring both sides et al...
I'm lazy
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May 15th, 2012, 09:55 AM   #5
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Assuming |x| ? 1, let x = sin(2A), where |2A| ? ?/2, and recall that cos(2A) = 2cos(A) - 1 = 1 - 2sin(A).

The equation becomes |sin(A)| + |cos(A)| = ?(1 + sin(2A)).

Squaring gives 1 + |sin(2A)| = 1 + sin(2A), i.e. |x| = x, so 0 ? x ? 1.
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