Question

Solve (x-5)³ - 27 = 0 for x, in the complex numbers domain.

a) x = 2
b) x = 5
c) x = 8
d) x = 10

Answer 1

To solve the equation (x-5)³ - 27 = 0 for x in the complex numbers domain, first simplify the equation by expanding the cube of (x-5). Next, factor out the common factor, (x-8), and apply the quadratic formula to find the roots of the quadratic factor. The correct answer is x = 8.

Step-by-step explanation:

To solve the equation (x-5)³ - 27 = 0 for x in the complex numbers domain, we can first simplify the equation by expanding the cube of (x-5). This gives us x³ - 15x² + 75x - 125 - 27 = 0. Combining like terms, we have x³ - 15x² + 75x - 152 = 0.

Next, we can factor out the common factor of (x-8) from the equation: (x-8)(x² - 7x + 19) = 0. By using the zero product property, we set each factor equal to zero and solve for x. The first factor gives us x-8 = 0, so x = 8. To solve the quadratic factor (x² - 7x + 19) = 0, we can apply the quadratic formula to find its roots.

Therefore, the correct answer is c) x = 8.