Question

use the discriminant to determine the number of solutions of -6x^2 - 108 = 0. which of the following statements gives the correct explanation?

Answer 1

To determine the number of solutions of the equation -6x^2 - 108 = 0 using the discriminant, we first need to identify the coefficients of the equation. In this case, the coefficient of x^2 is -6 and the constant term is -108.

The discriminant (Δ) is calculated as follows: Δ = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation (ax^2 + bx + c = 0).

For the equation -6x^2 - 108 = 0, a = -6, b = 0, and c = -108. Plugging these values into the discriminant formula, we get:

Δ = b^2 - 4ac = 0^2 - 4(-6)(-108)

Simplifying this expression gives us:

Δ = 0 - (-2592) = 2592

Now, the number of solutions can be determined based on the value of the discriminant according to the following cases:

1. If Δ > 0, then the equation has two distinct real solutions.
2. If Δ = 0, then the equation has one real solution (a double root).
3. If Δ < 0, then the equation has no real solutions.

In this case, Δ = 2592, which is greater than zero. Therefore, the correct statement that explains the number of solutions is:

"The equation -6x^2 - 108 = 0 has two distinct real solutions."