1

Whats so special about the identity $(a+b)^3 = a^3 + b^3 + 3ab(a+b)$ which gave Cardano the clever idea to invent a method to solve cubic depressed equations?

C. Ventin
  • 454
Orion_Pax
  • 431
  • 4
  • 12

1 Answers1

2

Assume that you have a depressed cubic equation, such as

$$x^3 + bx + c = 0.$$

After (apparent) centuries, Cardano (or somebody) got clever. They reasoned that you could set

$$x = (s + t)$$

and add a second constraint that $3st = -b.$

This meant that

$$s^3 + t^3 + 3st(x) + bx + c = 0. \tag1 $$

However, because of the artificial but valid constraint that $3st = -b$, equation (1) above implies that

$$s^3 + t^3 + c = 0. \tag2 $$

That is, because of this artifical constraint, the $x$ term has vanished.

Equation (2) above implies that

$$s^3 + \left[\frac{-b}{3s}\right]^3 + c = 0. \tag3 $$

In (3) above, multiplying each term by $(s^3)$ converts the equation into a quadratic equation in $(s^3).$

user2661923
  • 35,619
  • 3
  • 17
  • 39
  • Can this same /similar additional constraint term be applied to reduced quadratics/pentic/hextic equations? – Orion_Pax Apr 06 '22 at 06:30
  • @Orion_Pax My knowledge of Field Theory (especially Galois Theory) is practically non-existent. Once Cardano did his thing, over the next few centuries, something similar was done for quartic equations. Then, during this period, the professional Mathematicians thought that a similar method should work on quintic equations. In the late 1700's and early 1800's, it was discovered (i.e. proven) that this presumption is false. Personally, my understanding of the Math in this area is very hazy. The question that you are asking really involves Galois Theory. – user2661923 Apr 06 '22 at 06:36
  • @Orion_Pax I suspect that it would be quite a challenge for any Galois Theory guru to provide an explanation, in $500$ words or less of intuitively why there is no general solution to the quintic equation. Your comment/question has gone well beyond my knowledge. – user2661923 Apr 06 '22 at 06:38
  • I see its okay i have got a good idea how it was made useful like for cubic , i will try to search out for other degrees . Thanks for your wonderful explanation – Orion_Pax Apr 06 '22 at 06:49