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)

Question

Responses

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

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.

Answer 1

To determine the number of real solutions for the equation \(-6x^2 - 108 = 0\), we can start by rewriting it in the standard quadratic form \(ax^2 + bx + c = 0\). The equation simplifies to:

\[ -6x^2 = 108 \quad \Rightarrow \quad 6x^2 = -108 \quad \Rightarrow \quad x^2 = -18 \]

In this form, we identify the coefficients as:

• \(a = -6\)
• \(b = 0\)
• \(c = -108\)

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

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

Substituting in the values, we have:

\[ D = 0^2 - 4(-6)(-108) = 0 - 4(6)(108) = -2592 \]

Since the discriminant \(D\) is negative \((-2592 < 0)\), this implies that there are no real solutions to the equation.

Thus, the correct explanation is:

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