Final answer:
The equation 8z³+27=0 can be factored as a sum of cubes, leading to one real solution z = -1.5, as the quadratic factor yields no real roots.
Step-by-step explanation:
The equation to solve is 8z³+27=0. First, we can notice that this is a sum of cubes since 8z³ is (2z)³ and 27 is 3³. The sum of cubes can be factored as (a³ + b³) = (a + b)(a² - ab + b²). Using this pattern, we can rewrite our equation as (2z + 3)((2z)² - 2z*3 + 3²) = (2z + 3)(4z² - 6z + 9) = 0.
Setting each factor equal to zero gives us the possible solutions for z. From the linear factor (2z + 3) = 0, we can solve for z to get z = -3/2 or z = -1.5. However, the quadratic factor will not provide any real solutions since (4z² - 6z + 9) = 0 has a discriminant of b²-4ac that is negative (36 - 4*4*9 = -36), indicating no real roots.
Therefore, the only real solution to the equation is z = -1.5.