Final answer:
The equation z³ - 1 = 0 is solved by first recognizing it as a difference of cubes, then factoring and using the quadratic formula for the resulting quadratic equation to find one real and two complex solutions.
Step-by-step explanation:
To solve the equation z³ - 1 = 0, we can factor it as a difference of cubes because it can be written as z³ - 1³ = 0. The difference of cubes formula is a³ - b³ = (a - b)(a² + ab + b²). Using this formula, our equation factors to (z - 1)(z² + z + 1) = 0. From the first factor, we get a real solution z = 1. The second factor is a quadratic equation, and we use the quadratic formula to solve for the complex solutions.
The quadratic equation is in the form ax² + bx + c = 0. For our equation z² + z + 1 = 0, a = 1, b = 1, and c = 1. Applying the quadratic formula gives us:
z = [-b ± sqrt(b² - 4ac)] / (2a)
After substituting the values of a, b, and c, we find the two complex solutions for z. This gives us a total of three solutions for the original equation, including the real solution and two complex solutions.