Question

Question

The equation: x3=−8
will have how many solutions for x ?(1 point)
Responses

Two solutions: x = 2 and x = -2 since 2^3 = 8 and (-2)^3 = -8
Two solutions: x = 2 and x = -2 since 2^3 = 8 and (-2)^3 = -8

Two solutions: x = -8/3 and x = 8/3 since x^3 equations always have 2 solutions
Two solutions: x = -8/3 and x = 8/3 since x^3 equations always have 2 solutions

One solution: x = -2 since (-2)^3 = -8
One solution: x = -2 since (-2)^3 = -8

One solution: x = -8/3 since the inverse operation is to divide by 3.

Answer 1

The equation \( x^3 = -8 \) can be rewritten as \( x^3 + 8 = 0 \). This is a cubic equation and will have one real solution and two complex solutions.

To find the real solution, you can take the cube root of both sides:

\[ x = \sqrt[3]{-8} = -2 \]

This means that the only real solution is \( x = -2 \).

Regarding the complex solutions, they can be derived from the fact that every cubic equation has exactly three roots (real or complex). The remaining two roots could be derived using the cubic root of unity, but since your question focuses on the number of solutions, it's sufficient to say:

The equation \( x^3 = -8 \) has one real solution: \( x = -2 \).

Thus, the correct response is:

One solution: x = -2 since (-2)^3 = -8.