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:
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.
, 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:
And the formula for solving it has two solutions:
Well, applying it for:
, we have
Then, the <em>first solution</em> is:
The <em>second solution</em> is:
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:
Therefore
or
(rounding to four decimal places).