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### How is the golden ratio worked out?

(a+b)/a=a/b=phi (the golden ratio, as defined)
(a+b)/a=phi (we'll solve this equation)
1+b/a=phi (just changing the form of the left side a little)
1+1/phi=phi (a/b=phi so b/a=1/phi)
phi+1=phi2 (multiply both sides by phi)
phi2-phi-1=0 (rearrange)
From here, we can use the quadratic equation to find the
positive solution:
phi=(-b+√(b2-4ac))/(2a)
phi=(1+√(1+4))/2
phi=(1+√5)/2≈1.618