Final answer:
To solve the equation 8y^3 - 216 = 0, factor out the greatest common factor, recognize 27 as a perfect cube, set each factor equal to zero, and solve using the quadratic formula. The solution is y = 3, corresponding to option b.
Step-by-step explanation:
To solve the equation 8y^3 - 216 = 0, we can start by factoring out the greatest common factor, which is 8:
8(y^3 - 27) = 0
Next, we recognize that 27 is a perfect cube, so we can rewrite it as (3^3):
8(y - 3)(y^2 + 3y + 9) = 0
Setting each factor equal to zero, we have:
y - 3 = 0 --> y = 3
y^2 + 3y + 9 = 0
Using the quadratic formula, we find two complex solutions for y^2 + 3y + 9 = 0:
y = (-3 ± √(-3^2 - 4(1)(9)))/(2(1))
y = (-3 ± √(-3^2 - 36))/2
y = (-3 ± √(-45))/2
y = (-3 ± 3√5*i)/2
Therefore, the solutions to the equation are y = 3 or y = (-3 ± 3√5*i)/2. Since we are only considering real solutions, the correct answer is y = 3, which corresponds to option b.