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**Pn0yS0ld13r** Definition: If $\displaystyle \alpha$ is an irrational number in $\displaystyle \mathbf{Q}\left(\sqrt{d}\right)$, then the equation $\displaystyle ax^{2}+bx+c=0$ is called the defining equation for $\displaystyle \alpha$ if $\displaystyle \alpha$ satisfies the equation and a, b, and c are integers, $\displaystyle (a,b,c)=1$, and $\displaystyle a>0$.

Find a defining equation for the golden ratio $\displaystyle \dfrac{1+\sqrt{5}}{2}$.

How do you approach this problem?