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Thread: Defining equation for golden ratio; quadratic fields

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    Defining equation for golden ratio; quadratic fields

    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?
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    Hello,
    Quote Originally Posted by Pn0yS0ld13r View Post
    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?
    The solutions to such an equation are :

    $\displaystyle x=\frac{-b {\color{red}\pm} \sqrt{b^2-4ac}}{2a}$

    Since a and b are integers, the only way to get this $\displaystyle \sqrt{5}$ is $\displaystyle \sqrt{b^2-4ac}$.
    So you can see that $\displaystyle \frac{1{\color{red}-}\sqrt{5}}{2}$ will also be a root. (The reasoning may be similar to the ones involving complex roots).

    Develop $\displaystyle \left(x-\frac{1+\sqrt{5}}{2}\right)\left(x-\frac{1-\sqrt{5}}{2}\right)$ and identify a,b and c
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  3. #3
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    Thank you Moo.

    I can't believe I didn't get this before...
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