01dc4312-b146-4029-bc1c-b3cb1a4b574b
prompter
Hi, could you help me to solve this cubic equation using Cardano's Mehod (step by step if possible), please? -> " x^3 + 2x^2 - x - 1 = 0 "
33f2ea40-90a1-4a14-8ce1-1b6b40ee7637
01dc4312-b146-4029-bc1c-b3cb1a4b574b
assistant
Sure, I can help you solve the cubic equation x^3 + 2x^2 - x - 1 = 0 using Cardano's method. Here are the steps:

Step 1: Put the equation in the form x^3 + px + q = 0

x^3 + 2x^2 - x - 1 = 0 can be rearranged as x^3 + 0x^2 + 2x^2 - x - 1 = 0, so we have p = 0 and q = -1.

Step 2: Substitute x = y - b/3 in the equation, where b = -p/3

b = -p/3 = 0/3 = 0, so we have b = 0. Substituting x = y in the equation, we get:

(y - 0)^3 + 0(y - 0) - 1 = 0
y^3 - 1 = 0

Step 3: Solve for y

y^3 - 1 = 0 can be factored as (y - 1)(y^2 + y + 1) = 0. The roots of this equation are:

y = 1 and y = (-1 ± sqrt(3)i)/2

Step 4: Find the roots of the original equation

Substituting y = 1 into x = y - b/3, we get x = 1 - 0/3 = 1. So, 1 is one of the roots of the original equation.

Substituting y = (-1 + sqrt(3)i)/2 into x = y - b/3, we get:

x = (-1 + sqrt(3)i)/2 - 0/3 = (-1 + sqrt(3)i)/2

Substituting y = (-1 - sqrt(3)i)/2 into x = y - b/3, we get:

x = (-1 - sqrt(3)i)/2 - 0/3 = (-1 - sqrt(3)i)/2

So, the roots of the original equation are:

x = 1, x = (-1 + sqrt(3)i)/2, and x = (-1 - sqrt(3)i)/2

These are the three complex roots of the cubic equation x^3 + 2x^2 - x - 1 = 0.