How is the golden ratio worked out?

Answer 1

(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