- #1
cscott0001
- 6
- 1
Homework Statement
Use Cardano's formula to find a real root for ##3x^3-45x^2+243x-525=0##. [Edited to correct mistake]
Homework Equations
$$x = u - \frac{b}{3a}$$
Depressed cubic: $$u^3=3pu+2q$$
Cardano's formula: $$u=\sqrt[3]{q+\sqrt{q^2-p^3}}+\sqrt[3]{q-\sqrt{q^2-p^3}}$$
The Attempt at a Solution
I have found the depressed cubic to be ##x^3=20-6x##. I found ##3p= -6## and ##2q=20 \to p=-2## and ##q=10##. Using Cardano's formula, I arrived at ##u=\sqrt[3]{10+\sqrt{108}}+\sqrt[3]{10-\sqrt{108}}##. However, I graphed the equation and know that it should be 7. If I change my answer to ##u=\sqrt[3]{10+\sqrt{108}}-\sqrt[3]{-10+\sqrt{108}}## and use this answer (2) to solve ##x = u - \frac{b}{3a}## I get the right answer (7), but I have no idea mathematically why I would do that, or how I'd arrive there. I have been staring at this for two days with little progress; where did I go wrong?
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