The tutor introduces the golden ratio.

The golden ratio is defined as x/y, where

x/y = (x+y)/x, x>y, y>0

Muliplying both sides by xy gives

x^2=xy+y^2

which leads to

x^2-xy-y^2=0

Using the quadratic formula to solve for x, we imagine

x=(-b ± (b^2-4ac)^0.5)/(2a)

based on

ax^2+bx+c=0

In our particular equation is

a=1, b=-y, c=-y^2

which leads to

x=(-(-y)±((-y)^2-4(1)(-y^2))^0.5)/(2*1)

Simplifying, we get to

x=(y+(5y^2)^0.5)/2 or x=(y-(5y^2)^0.5)/2

which becomes

x=(y+y(5)^0.5)/2 or (y-y(5)^0.5)/2

Since x>0,

x=(y+y(5)^0.5)/2 only.

Continuing to simplify, we get

x=y(1+5^0.5)/2

So the golden ratio, x/y, is

x/y = (y(1+5^0.5)/2)/y = (1+5^0.5)/2 ≈ 1.618

I’ll be discussing the golden ratio further in future posts:)

Source:

Wikipedia

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.