Use the discriminant to determine the number of real solutions of −6x2−108=0 . Which of the following statements gives the correct explanation?(1 point) Responses There will be two real solutions since the discriminant is positive. There will be two real solutions since the discriminant is positive. There will be one real solution since the discriminant is zero. There will be one real solution since the discriminant is zero. There will be no real solutions since the discriminant is zero. There will be no real solutions since the discriminant is zero. There will be no real solutions since the discriminant is negative.

Answer 1

To determine the number of real solutions of the quadratic equation \(-6x^2 - 108 = 0\), we can rewrite it in the standard form \(ax^2 + bx + c = 0\) where \(a = -6\), \(b = 0\), and \(c = -108\).

The discriminant \(D\) of a quadratic equation is given by the formula:

\[ D = b^2 - 4ac \]

Substituting in the values of \(a\), \(b\), and \(c\):

\[ D = 0^2 - 4(-6)(-108) = 0 - 4 \cdot 6 \cdot 108 \]

Calculating \(4 \cdot 6 \cdot 108\):

\[ 4 \cdot 6 = 24 \quad \text{and} \quad 24 \cdot 108 = 2592 \]

Thus, the discriminant is:

\[ D = 0 - 2592 = -2592 \]

Since the discriminant is negative, this indicates that there are no real solutions to the equation.

Therefore, the correct response is:

There will be no real solutions since the discriminant is negative.