<s>[INST] How do you factor $x^2-x-1$? I know you can't have all integers, but how do you factor this anyway?
Wolfram|Alpha gives me $-\frac{1}{4} (1+\sqrt{5}-2 x) (-1+\sqrt{5}+2 x)$.
Cymath gives me $(x-\frac{1+\sqrt{5}}{2})(x-\frac{1-\sqrt{5}}{2})$.
The closest I can get is $(x+1)(x-1)-x$.
So how do I get a nice answer like the ones listed above?
 [/INST] Apply quadratic formula for the roots of $x^2-x-1=0$ as follows 
$$x=\frac{-(-1)\pm\sqrt{(-1)^2-4(1)(-1)}}{2(1)}=\frac{1\pm \sqrt 5}{2}$$
hence, one should have the following factors  $$x^2-x-1=1\cdot \left(x-\frac{1+ \sqrt 5}{2}\right)\left(x-\frac{1- \sqrt 5}{2}\right)$$
or $$\frac{1}{4}(2x-1-\sqrt 5)(2x-1+\sqrt 5)$$
$$=-\frac{1}{4}(1+\sqrt 5-2x)(-1+\sqrt 5+2x)$$
So both the answers are correct
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