Golden ratio is represented by the symbol ϕ(Phi), and its conjugate is –ϕ(phi, also called as silver ratio). Both are satisfied by the equation, $x^2 – x – 1 = 0$, as given in the explanation for Golden ratio. Since this equation is a quadratic equation, we are going to solve it the usual way, by calculating its roots.
Roots $= \frac{-b \pm \sqrt {b^2-4ac}}{2a}.$
Given the equation, $x^2 – x – 1 = 0$, value of $a = 1,b = -1$ and $c = -1$. So, the equation for calculating root will become
$\frac{1 \pm \sqrt{1+4}} { 2 }$
$= \frac{1 \pm \sqrt(5)}{2}$
If we calculate the roots we will get the value 1.61 and -0.61 which are actually the values of Golden ratio and its conjugate. So, the Golden ratio and its conjugate both satisfy the equation $x^2 – x – 1 = 0$. The answer is C.
If you want to know more then, Here is the reference. He has explained very beautifully. Have a look.
