<s>[INST] Proving the second root of a quadratic equation If $\alpha$ is a root of the equation $4x^2+2x-1=0$, then prove that $4\alpha^3-3\alpha$ is the other root. How do I proceed? The sum of the roots, the product of the roots lead me nowhere. Should I find the roots of the equation and substitute in the given two expressions of $\alpha$ and check whether manipulating the first root gives me the second root (which seems much complicated), or is there any other way?
 [/INST] Clearly, the absolute value of both roots are $<1$
$$0=(4x^2+2x-1)(x-1)=4x^3-3x-(2x^2-1)$$
If $x=\cos y,$ we have $$\cos3y=\cos2y$$
$\implies3y=360^\circ n\pm2y$ where $n$ is any integer.
'+'$\implies y=360^\circ n\implies x=1$
'-'$\implies y=72^\circ n$ where $n\equiv0,\pm1,\pm2\pmod5$
As $x=1$  does not satisfy $4x^2+2x-1=0,n\not\equiv0\pmod5$
So, the roots of $4x^2+2x-1=0$ are $\cos72^\circ,\cos144^\circ=-\cos36^\circ$
Now $4\cos^372^\circ-3\cos72^\circ=\cos3\cdot72^\circ=\cos216^\circ=\cdots=-\cos36^\circ$
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Similarly, $\cos3\cdot(144^\circ)=\cdots=\cos72^\circ$
Can you take it from here
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