Fibonacci's proof that $x^3+2x^2+10x=20$ has no solution in radicals?

Question

I read on a poster today that Fibonacci showed that $x^3+2x^2+10x=20$ has no solution expressible in radicals, way back when.

I couldn't find the proof anywhere. Does anyone know where I can find it?

Answer 1

Your claim is wrong. Here is the one real root the polynomial has. Define $\alpha = \sqrt[3]{176+3\sqrt{3930}}$. Then the real root is $$\dfrac13\left(-2 - \dfrac{13 \cdot 2^{2/3}}{\alpha} + \alpha\sqrt[3]2\right)$$