How to Show Phi Is Quadratic

Step-by-Step Guide

1. 1

   Step 1: Open a new workbook in Excel and do (list in text) the following algebraic steps (reset column width to accommodate): Do a/b = b/(a+b) Do a(a+b) = b^2 ... where "b^2" denotes b squared.

   Do a^2 + ab
   - b^2 = 0 Let a = x, b=b, c=b^2 Do x^2 + bx
   - c = 0 Let a=1 Let ax^2 + bx
   - c = 0; this is a standard quadratic form.
   
   Then let the quadratic roots be given by ... {x1, x2} = (-b ± sqrt(b^2
   - 4ac)/ 2a Let c=-b^2 and let a=1 Let {x1, x2} = (-b ± sqrt(b^2
   - 4*1*(-b^2))/ (2*1) Let {x1, x2} = (-b ± sqrt(5b^2)/ 2 Let {x1, x2} = {Phi1, Phi2} and let b = 1 Let {Phi1, Phi2} = (-1 ± sqrt(5*(1^2)))/2 Let {Phi1, Phi2} = (-1 ± sqrt(5))/2; Done!
2. 2

   Step 2: Compute the results of the formulas: Let Phi1 = (-1 + sqrt(5)) / 2 = 0.618033988749895

   Let Phi2 = (-1
   - sqrt(5)) / 2 =
   -1.61803398874989 , Let Phi2 = (--1
   - sqrt(5)) / 2 =
   -0.618033988749895
3. 3

   Step 3: which is the artistic ratio for Phi.

4. 4

   Step 4: Please note that had we let b=-1

5. 5

   Step 5: the results would have been: Let Phi1 = (--1 + sqrt(5)) / 2 = 1.618033988749895

6. 6

   Step 6: which = Phi

7. 7

   Step 7: formally in Math.

Detailed Guide

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