Solving cubic equation using Cardano's method

In my university level history of mathematics course I have encountered the following problem, I have to admit I am stuck!

Solve the following cubic function using Cardanos method: x^{3} + 3x = 2

(Cardano, Ars magna, Caput XI, De Cubo & rebus aequalibus Numero.)

Also calculate an approximation with three decimals (using the method)

Any help would be much appreciated!

Re: Solving cubic equation using Cardano's method

Do you know what "Cardano's method" **is**? If you do, it should be a matter of just doing the arthmetic!

If a and b are any real numbers, the $\displaystyle (a- b)^3= a^3- 3a^2b+ 3ab^2- b^3$ and $\displaystyle 3ab(a- b)= 3a^2b- 3ab^2$ so that $\displaystyle (a- b)^3+ 3ab(a- b)= a^3- b^3$.

So if we let $\displaystyle x= a- b$, $\displaystyle m= 3ab$, and $\displaystyle n= a^3- b^3$, we have $\displaystyle x^3- mx= n$. Now, the question is "If we know m and n, can we find a and b?" (and so x).

From $\displaystyle m= 3ab$, we have $\displaystyle b= \frac{m}{3a}$. Putting that into $\displaystyle a^3- b^3= n$, we have $\displaystyle a^2- \frac{m^3}{3^3a^3}= n$ and, multiplying through by $\displaystyle a^3$, $\displaystyle (a^3)^2- \left(\frac{m}{3}\right)^3= na^3$ which is equivalent to the quadratic equation $\displaystyle (a^3)^3- n(a^3)+ \left(\frac{m}{3}\right)^3= 0$ for $\displaystyle a^3$.

We can use the quadratic equation to solve that for $\displaystyle a^3$: $\displaystyle a^3= \frac{n\pm\sqrt{n^2- 4\left(\frac{m}{3}\right)^3}}{2}= \frac{n}{2}\pm\sqrt{\left(\frac{n}{2}\right)^2- \left(\frac{m}{3}\right)^3}$.

From $\displaystyle a^3- b^3= n$, $\displaystyle b^3= a^3- n= -\frac{n}{2}\pm\sqrt{\left(\frac{n}{2}\right)^2- \left(\frac{m}{3}\right)^3}$.

Once you have determined $\displaystyle a^3$ and $\displaystyle b^3$, take the cube roots to find a and b and then find x= a- b. Of course, you will need to be careful about the three cube roots (two will be complex and one real).

Re: Solving cubic equation using Cardano's method

I do know what it is, but I did not have the skills to solve it! Thank you very much

Re: Solving cubic equation using Cardano's method

I just have a homework question I've been stuck on and hoping you guys can help me out a bit.

COnsider the number 48. If you add 1 to it, you get 49, which is a perfect square. If you add 1 to its (1/2), you get 25, which is also a perfect square. Please find the next 2 numbers with the same properties. like 48+1=49 (perfect square)

48/2=24, 24+1=25 (perfect square)

Thank you so much!