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Three expressions - same value (Posted on 2016-06-02) Difficulty: 3 of 5
For a certain couple of two positive numbers a and b the sum a+b, the product a*b and the expression a^2-b^2 result as the same non-zero number.

Evaluate a and b .

No Solution Yet Submitted by Ady TZIDON    
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Solution computer assisted solution | Comment 1 of 3
ab = a + b
b(a-1) = a
b = a/(a-1)

a^2 - a^2/(a-1)^2 = a + a/(a-1) 

After graphing a^2 - a^2/(a-1)^2 - (a + a/(a-1) ) and finding its zeros at (a) zero, (b) between zero and 1, and (c) between 2 and 3, I narrowed down the search with Basic computer programs seen below.  A graphing calculator would accomplish the same purpose.

DEFDBL A-Z
OPEN "3eqexp.txt" FOR OUTPUT AS #2
CLS
FOR a = .381966011250105# TO .381966011250106# STEP .0000000000000001#
     PRINT a, a ^ 2 - a ^ 2 / (a - 1) ^ 2 - (a + a / (a - 1))
     PRINT #2, a, a ^ 2 - a ^ 2 / (a - 1) ^ 2 - (a + a / (a - 1))
NEXT
CLOSE

        A                  difference between first and last expressions
                             assuming A+B = A*B
 .381966011250105            1.412715430748612D-16 
 .3819660112501051           4.639030045522352D-17    3/2 - sqrt(5)/2
 .3819660112501052          -4.846383711010205D-17 
 .3819660112501053          -1.432366595124912D-16 
 .3819660112501054          -2.380501394963486D-16 
 .3819660112501055          -3.329042770616741D-16 
 .3819660112501057          -4.277584146269997D-16 
 .3819660112501058          -5.225854471380131D-16 
 .3819660112501059          -6.173989271218705D-16 
 .381966011250106           -7.121853020514157D-16 
 
However in this result, B is negative at -.618033988749895.



DEFDBL A-Z
OPEN "3eqexp.txt" FOR OUTPUT AS #2
CLS
FOR a = 2.618033988749# TO 2.61803398875# STEP .0000000000001#
     PRINT a, a ^ 2 - a ^ 2 / (a - 1) ^ 2 - (a + a / (a - 1))
     PRINT #2, a, a ^ 2 - a ^ 2 / (a - 1) ^ 2 - (a + a / (a - 1))
NEXT
CLOSE

 2.618033988749             -5.238165803889139D-12 
 2.6180339887491            -4.653222685641056D-12 
 2.6180339887492            -4.068280434754712D-12 
 2.6180339887493            -3.483338617549236D-12 
 2.6180339887494            -2.898395932982023D-12 
 2.6180339887495            -2.313454115776548D-12 
 2.6180339887496            -1.728511431209334D-12 
 2.6180339887497            -1.143569614003859D-12 
 2.618033988749799          -5.586269294366453D-13  1 + phi
 2.618033988749899           2.631488776883018D-14 
 2.618033988749999           6.112584396977816D-13 

Here B is positive, equal to phi, the golden mean.

Answers: A = 1 + phi, B = phi

In both programs, the version shown is after narrowing down the values one decimal place at a time.

  Posted by Charlie on 2016-06-02 15:25:05
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