Solving cubic equation using Cardano's method

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May 21st 2012, 01:11 AM
glutto

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: x³ + 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!
May 22nd 2012, 10:57 AM
HallsofIvy

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 $(a- b)^3= a^3- 3a^2b+ 3ab^2- b^3$ and $3ab(a- b)= 3a^2b- 3ab^2$ so that $(a- b)^3+ 3ab(a- b)= a^3- b^3$ .

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

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

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

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

Once you have determined $a^3$ and $b^3$ , 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).
May 24th 2012, 01:20 AM
glutto

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
May 24th 2012, 04:57 PM
themathlete1

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!