# How to check equations/identities?

So, i got a different cubic formula for the depressed equation than that was there in the standard formula from a book. $y^3+fy+g=0$ has the solution

$({-g+\sqrt {g^2/4+f^3/27} })^{1/3} + ({-g-\sqrt {g^2/4+f^3/27} })^{1/3}$

$-\frac{f}{3 (-\frac{1}{2}g + \frac{1}{6} \sqrt{\frac{4}{3}f^3+9g^2}) ^{1/3} } == ({-g+\sqrt {g^2/4+f^3/27} })^{1/3}$

How do I check. I want to use sage,not math. In particular the bool() function.

edit retag close merge delete

Sort by » oldest newest most voted

Code:

sage: var( 'f,g' );
sage: LHS = - f/3/ ( -g/2 + 1/6*sqrt( 4/3*f^3 + 9*g^2) )^(1/3)
sage: RHS =        ( -g/2 - 1/6*sqrt( 4/3*f^3 + 9*g^2) )^(1/3)
sage: bool( LHS == RHS )
False
sage: bool( LHS^3 == RHS^3 )
True


If a bool on a to be equality is answered by sage with False, this means that the equality is mathematically either false (for instance if we mistype a g/2 as g or a plus as a minus in the formulas for LHS or RHS above), or true, but sage could not prove it in the latter case. In case sage gives us the True, then it is really true in mathematics. In our case, sage is giving us the True only after passing to the third power. (And i would do the same mathematically, since without doing this it depends on the implicit choice of the third root for the two places to make the choice. If one correlates the roots, this is ok. But where should sage or i know this?)

more