What are the real and complex solutions of the polynomial equation?
x^3-64=0
The polynomial equation x^3 - 64 = 0 can be factored as (x - 4)(x^2 + 4x + 16) = 0.
The real solution is x = 4.
The complex solutions can be found by solving the quadratic equation x^2 + 4x + 16 = 0 using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = 1, b = 4, and c = 16.
x = (-4 ± √(4^2 - 4(1)(16))) / (2(1))
x = (-4 ± √(16 - 64)) / 2
x = (-4 ± √(-48)) / 2
x = (-4 ± 4√(-3)) / 2
x = -2 ± 2√(-3)
So, the complex solutions are x = -2 + 2√(-3) and x = -2 - 2√(-3).