I was given the following simultaneous equations to solve on a homework sheet:
$$
x^2 + y^2 = 3\\x-y=1
$$
And when I did so I got the answers of:
$$
(\varphi,-1/\varphi)\\
(1/\varphi,-\varphi)
$$
I checked the answer sheet, and this was the correct answer. I know how this is correct, having done the quadratic formula and all to work it out in the first place, but what I don't get is why. The first equation, $$x^2 + y^2 = 3$$ plots a circle of radius root 3, yet the point $$(\varphi,-1/\varphi)$$ is also on this circle. By my knowledge of alegebra and pythagoras, this hence implies that
$$ \varphi^2 + -1/\varphi^2 = 3 $$
However, given that the golden ratio is
$$ \frac{\sqrt{5}+1}{2} $$
I fail to see how or where a 3 or a root 3 can come from.
Could someone explain how you get from the golden ratio, a very root-5-y constant, to root 3?
Best Answer
We have that $$x^2+\frac1{x^2}=3 \iff x^4-3x^2+1=0 \iff x^2=\frac{3\pm\sqrt{5}}{2} $$
and then by nested square roots
$$x_{1,2}=\pm \sqrt{\frac{3+\sqrt{5}}{2}}=\pm \frac{1+\sqrt 5}{2}$$
$$x_{3,4}=\pm \sqrt{\frac{3-\sqrt{5}}{2}}=\pm \frac{-1+\sqrt 5}{2}$$
and indeed
$$x^4-3x^2+1=(x^2+x-1)(x^2-x-1)$$