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 .
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.
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Posted by Charlie
on 2016-06-02 15:25:05 |
computer assisted solution
