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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

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

$\left{y : -\frac{f}{3 \, {\left(-\frac{1}{2} \, g + \frac{1}{6} \, \sqrt{\frac{4}{3} \, f^{3} + 9 \, g^{2}}\right)}^{\frac{1}{3}}} + {\left(-\frac{1}{2} \, g + \frac{1}{6} \, \sqrt{\frac{4}{3} \, f^{3} + 9 \, g^{2}}\right)}^{\frac{1}{3}}\right}$

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

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

$\left{y : -\frac{f}{3 \, {\left(-\frac{1}{2} \, g + \frac{1}{6} \, \sqrt{\frac{4}{3} \, f^{3} + 9 \, g^{2}}\right)}^{\frac{1}{3}}} + {\left(-\frac{1}{2} \, g + \frac{1}{6} \, \sqrt{\frac{4}{3} \, f^{3} + 9 \, g^{2}}\right)}^{\frac{1}{3}}\right}$

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

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

$ \left{y \left\{y : -\frac{f}{3 \, {\left(-\frac{1}{2} \, g + \frac{1}{6} \, \sqrt{\frac{4}{3} \, f^{3} + 9 \, g^{2}}\right)}^{\frac{1}{3}}} + {\left(-\frac{1}{2} \, g + \frac{1}{6} \, \sqrt{\frac{4}{3} \, f^{3} + 9 \, g^{2}}\right)}^{\frac{1}{3}}\right} g^{2}}\right)}^{\frac{1}{3}}\right\} $

So I want to check the algebraic equivalence of these two equations step by step, and how I can convert the first equation into the second, as they are both the same.

This is like checking algebra formulas like $(a+b)^2=a^2+b^2+2ab$$(a+b)^2=a^2+b^2+2ab$ If nothing else, i want to just check True or False, just like with numbers, but equations instead.

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

$\left\{y : -\frac{f}{3 \, {\left(-\frac{1}{2} \, g + \frac{1}{6} \, \sqrt{\frac{4}{3} \, f^{3} + 9 \, g^{2}}\right)}^{\frac{1}{3}}} + {\left(-\frac{1}{2} \, g + \frac{1}{6} \, \sqrt{\frac{4}{3} \, f^{3} + 9 \, g^{2}}\right)}^{\frac{1}{3}}\right\}$

So I want to check the algebraic equivalence of these two equations step by step, and how I can convert the first equation into the second, as they are both the same.

This is like checking algebra formulas like $(a+b)^2=a^2+b^2+2ab$ If nothing else, i want to just check True or False, just like with numbers, but equations instead.