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 and so that .

So if we let , , and , we have . Now, the question is "If we know m and n, can we find a and b?" (and so x).

From , we have . Putting that into , we have and, multiplying through by , which is equivalent to the quadratic equation for .

We can use the quadratic equation to solve that for : .

From , .

Once you have determined and , 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!