Answer:
(rounding to four decimal places)
Step-by-step explanation:
<em>As a clarification note:</em> The Golden Ratio is usually represented by the Greek letter <em>phi</em> (
) and not by <em>rho</em> letter (
). To answer the question, we will use the <em>rho</em> letter.
To solve this equation:
![\\ \rho = 1 + \frac{1}{\rho}](https://tex.z-dn.net/?f=%20%5C%5C%20%5Crho%20%3D%201%20%2B%20%5Cfrac%7B1%7D%7B%5Crho%7D%20)
We need first to <em>rearrange</em> its terms and then apply the <em>quadratic formula</em> for it, since the result of such rearrangement is a <em>quadratic equation</em>.
<h3>Rearranging the formula</h3>
Then, the equation
can be multiplied by
to both of its sides. The equation remains the same in doing so.
![\\ \rho * (\rho = 1 + \frac{1}{\rho})](https://tex.z-dn.net/?f=%20%5C%5C%20%5Crho%20%2A%20%28%5Crho%20%3D%201%20%2B%20%5Cfrac%7B1%7D%7B%5Crho%7D%29%20)
![\\ \rho * \rho = \rho + \rho * \frac{1}{\rho}](https://tex.z-dn.net/?f=%20%5C%5C%20%5Crho%20%2A%20%5Crho%20%3D%20%5Crho%20%2B%20%5Crho%20%2A%20%5Cfrac%7B1%7D%7B%5Crho%7D%20)
![\\ {\rho}^2 = \rho + \frac{\rho * 1}{\rho}](https://tex.z-dn.net/?f=%20%5C%5C%20%7B%5Crho%7D%5E2%20%3D%20%5Crho%20%2B%20%5Cfrac%7B%5Crho%20%2A%201%7D%7B%5Crho%7D%20)
![\\ {\rho}^2 = \rho + \frac{\rho}{\rho} * 1](https://tex.z-dn.net/?f=%20%5C%5C%20%7B%5Crho%7D%5E2%20%3D%20%5Crho%20%2B%20%5Cfrac%7B%5Crho%7D%7B%5Crho%7D%20%2A%201%20)
![\\ {\rho}^2 = \rho + 1 * 1](https://tex.z-dn.net/?f=%20%5C%5C%20%7B%5Crho%7D%5E2%20%3D%20%5Crho%20%2B%201%20%2A%201%20)
![\\ {\rho}^2 = \rho + 1](https://tex.z-dn.net/?f=%20%5C%5C%20%7B%5Crho%7D%5E2%20%3D%20%5Crho%20%2B%201%20)
, which is a <em>quadratic equation</em> that can be solved using the well known <em>quadratic formula</em> aforementioned.
<h3>Solutions for the resulting equation</h3>
A quadratic equation is of the form:
![\\ a*x^2 + b*x + c = 0](https://tex.z-dn.net/?f=%20%5C%5C%20a%2Ax%5E2%20%2B%20b%2Ax%20%2B%20c%20%3D%200%20)
And the formula for solving it has two solutions:
![\\ x_{1} = \frac{-b + \sqrt{b^2 - 4*a*c}}{2*a}](https://tex.z-dn.net/?f=%20%5C%5C%20x_%7B1%7D%20%3D%20%5Cfrac%7B-b%20%2B%20%5Csqrt%7Bb%5E2%20-%204%2Aa%2Ac%7D%7D%7B2%2Aa%7D%20)
![\\ x_{2} = \frac{-b - \sqrt{b^2 - 4*a*c}}{2*a}](https://tex.z-dn.net/?f=%20%5C%5C%20x_%7B2%7D%20%3D%20%5Cfrac%7B-b%20-%20%5Csqrt%7Bb%5E2%20-%204%2Aa%2Ac%7D%7D%7B2%2Aa%7D%20)
Well, applying it for:
, we have ![a = 1, b = -1, c = -1](https://tex.z-dn.net/?f=%20a%20%3D%201%2C%20b%20%3D%20-1%2C%20c%20%3D%20-1%20)
Then, the <em>first solution</em> is:
![\\ \rho_{1} = \frac{-(-1) + \sqrt{(-1)^2 - 4*1*(-1)}}{2*1}](https://tex.z-dn.net/?f=%20%5C%5C%20%5Crho_%7B1%7D%20%3D%20%5Cfrac%7B-%28-1%29%20%2B%20%5Csqrt%7B%28-1%29%5E2%20-%204%2A1%2A%28-1%29%7D%7D%7B2%2A1%7D%20)
![\\ \rho_{1} = \frac{1 + \sqrt{(-1)(-1) - 4*1*(-1)}}{2*1}](https://tex.z-dn.net/?f=%20%5C%5C%20%5Crho_%7B1%7D%20%3D%20%5Cfrac%7B1%20%2B%20%5Csqrt%7B%28-1%29%28-1%29%20-%204%2A1%2A%28-1%29%7D%7D%7B2%2A1%7D%20)
![\\ \rho_{1} = \frac{1 + \sqrt{1 + 4}}{2*1}](https://tex.z-dn.net/?f=%20%5C%5C%20%5Crho_%7B1%7D%20%3D%20%5Cfrac%7B1%20%2B%20%5Csqrt%7B1%20%2B%204%7D%7D%7B2%2A1%7D%20)
The <em>second solution</em> is:
![\\ \rho_{2} = \frac{-(-1) - \sqrt{(-1)^2 - 4*1*(-1)}}{2*1}](https://tex.z-dn.net/?f=%20%5C%5C%20%5Crho_%7B2%7D%20%3D%20%5Cfrac%7B-%28-1%29%20-%20%5Csqrt%7B%28-1%29%5E2%20-%204%2A1%2A%28-1%29%7D%7D%7B2%2A1%7D%20)
![\\ \rho_{2} = \frac{1 - \sqrt{1 + 4}}{2} = \frac{1 - \sqrt{5}}{2}](https://tex.z-dn.net/?f=%20%5C%5C%20%5Crho_%7B2%7D%20%3D%20%5Cfrac%7B1%20-%20%5Csqrt%7B1%20%2B%204%7D%7D%7B2%7D%20%3D%20%5Cfrac%7B1%20-%20%5Csqrt%7B5%7D%7D%7B2%7D%20)
But, the Golden Ratio is a <em>positive number</em> since it is a ratio between <em>positive</em> quantities, thus, the valid solution for the Golden Ratio is the first solution. The second solution is a negative number.
, which is the exact solution.
Since the question is asking to <em>round this answer to four decimal places</em>, then we have:
![\sqrt{5} = 2.2360679774997896964091...](https://tex.z-dn.net/?f=%20%5Csqrt%7B5%7D%20%3D%202.2360679774997896964091...%20)
Therefore
![\\ \rho_{1} = \frac{1 + 2.2360679774997896964091...}{2}](https://tex.z-dn.net/?f=%20%5C%5C%20%5Crho_%7B1%7D%20%3D%20%5Cfrac%7B1%20%2B%202.2360679774997896964091...%7D%7B2%7D%20)
or
(rounding to four decimal places).