The golden ratio is a special number approximately equal to 1.618 that appears often in mathematics,

Question

geometry, art, architecture and other areas.
Given that the numbers 10 and x are in a golden ratio and that x is larger than 10, determine the exact
value of x. Only algebraic solutions will get full marks.

Answer 1

if a>b
then works if
(a+b) / a = a / b
here a = x and b = 10
so
(x+10) /x = x / 10
10 (x+10) = x^2
10 x + 100 = x^2
x^2 - 10 x -100 = 0
x = [ 10 +/- sqrt(100 + 400) ] / 2
use the + sign because x must be >10
x = [ 10 + sqrt 500 ] / 2 = (1/2) (10 + 10 sqrt 5) = 5 ( 1+sqrt 5)

Answer 2

sure enough, x is about 16.18033989