SOLUTION: I'm trying to prove that {{{sqrt(4 - 2 * sqrt(3)) + sqrt(12 - 6 * sqrt(3)) = 2}}}

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Question 681800: I'm trying to prove that
sqrt%284+-+2+%2A+sqrt%283%29%29+%2B+sqrt%2812+-+6+%2A+sqrt%283%29%29+=+2
Found 2 solutions by Alan3354, MathTherapy:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%284+-+2+%2A+sqrt%283%29%29+%2B+sqrt%2812+-+6+%2A+sqrt%283%29%29+=+2
2sqrt%281+-+0.5sqrt%283%29%29+%2B+sqrt%2812%29%2Asqrt%281+-+0.5sqrt%283%29%29+=+2
%282+%2B+2sqrt%283%29%29%2Asqrt%281+-+0.5sqrt%283%29%29+=+2
%281+%2B+sqrt%283%29%29%2Asqrt%281+-+0.5sqrt%283%29%29+=+1
Multiply by sqrt(1 + 0.5sqrt(3))
%281+%2B+sqrt%283%29%29%2Asqrt%281+-+0.25%2A3%29+=+sqrt%281+%2B+0.5sqrt%283%29%29
square both sides
%281+%2B+3+%2B+2sqrt%283%29%29%2A%280.25%29+=+1+%2B+0.5sqrt%283%29
1+%2B+0.5sqrt%283%29+=+1+%2B+0.5sqrt%283%29
QED

Answer by MathTherapy(10801) About Me  (Show Source):
You can put this solution on YOUR website!
I'm trying to prove that

sqrt%284+-+2+%2A+sqrt%283%29%29+%2B+sqrt%2812+-+6+%2A+sqrt%283%29%29+=+2            
*******************
This author's proof is different from the one the other person who responded, provided. His method
may not be what the person who needs help, is looking for. Then again, this hunch may be WRONG!

Having said that, let's focus on the L.H.S. because that's what's needed to be proven to equal 2.
                        sqrt%284+-+2sqrt%283%29%29 + sqrt%2812+-+6sqrt%283%29%29
                        sqrt%284+-+2sqrt%283%29%29 + sqrt%2812+-+2%283%29sqrt%283%29%29 ---- Changing 6 to 2(3)
                        sqrt%284+-+2sqrt%283%29%29 + sqrt%2812+-+2sqrt%289%29sqrt%283%29%29 ------- Changing 3 to sqrt%289%29 
               sqrt%283+%2B+1+-+2sqrt%283%29sqrt%281%29%29 + sqrt%289+%2B+3+-+2sqrt%289%29sqrt%283%29%29 ----- Changing 4 to 3 + 1, 3 to sqrt%283%29sqrt%281%29, and 12 to 9 + 3                 
sqrt%28%28sqrt%283%29%29%5E2+%2B+%28sqrt%281%29%29%5E2+-+2sqrt%283%29sqrt%281%29%29 + sqrt%28%28sqrt%289%29%29%5E2+%2B+%28sqrt%283%29%29%5E2+-+2sqrt%289%29sqrt%283%29%29 ------ Converting 
The above is in the form:
                          %28a%5B1%5D+-+b%5B1%5D%29%5E2  + %28a%5B2%5D+-+b%5B2%5D%29%5E2, with , and so:
sqrt%28%28sqrt%283%29%29%5E2+%2B+%28sqrt%281%29%29%5E2+-+2sqrt%283%29sqrt%281%29%29 + sqrt%28%28sqrt%289%29%29%5E2+%2B+%28sqrt%283%29%29%5E2+-+2sqrt%289%29sqrt%283%29%29 then becomes:
                   sqrt%28%28sqrt%283%29+-+sqrt%281%29%29%5E2%29 +  sqrt%28%28sqrt%289%29+-+sqrt%283%29%29%5E2%29
                         sqrt%283%29+-+sqrt%281%29    +    sqrt%289%29+-+sqrt%283%29 ----- Cancelling SQUARE and SQUARE ROOT
                             sqrt%283%29+-+1 + 3+-+sqrt%283%29+=+highlight%282%29%29QED!