\displaystyle x^2-x-1=0 \ \ ,\mathrm{aka\ the\ golden\ ratio\ equation\ ,\ it\ has\ solutions} \\ \\ \ x=\frac{1}{2}(1+ \sqrt 5)=1.6180...=\phi \ \mathrm{and} \ x=\frac{1}{2}(1- \sqrt 5)=-0.6180...=\psi