Question

an equation is shown.
$x^3 = -27$
what is true when solving the equation?
a. there are no real solutions to the equation because of the negative.
b. there is a positive and a negative solution.
c. there is only a negative solution to the equation.
d. there is only a positive solution to the equation.

Explanation:

To solve \(x^{3}=-27\), we find the cube root of both sides. The cube root of \(-27\) is \(-3\) because \((-3)^{3}=(-3)\times(-3)\times(-3)= -27\). For cubic equations \(x^{3}=a\), if \(a\) is negative, the real - valued solution is negative (since the cube of a negative number is negative, the cube of a positive number is positive, and the cube of 0 is 0). So, the equation \(x^{3}=-27\) has only one real solution, and that solution is negative.

• Option A is wrong because we found a real solution (\(x = - 3\)).
• Option B is wrong because there is only one real solution (not a positive and a negative).
• Option D is wrong because the solution is negative, not positive.

Answer:

C. There is only a negative solution to the equation.