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

There will be one real solution since the discriminant is zero.

Answer 1

To determine the number of real solutions for the equation \(-6x^2 - 108 = 0\), we can first rewrite it in standard form.

1. Rearranging the equation gives: \[ -6x^2 = 108 \] \[ x^2 = -18 \]

Now we can analyze the solutions based on the value of \(x^2\). Since \(x^2\) cannot be negative in the real number system, this means there are no real solutions.

Alternatively, we can find the discriminant using the quadratic formula for a general quadratic equation of the form \(ax^2 + bx + c = 0\). Here:

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

The discriminant \(D\) is given by: \[ D = b^2 - 4ac \] Substituting the values: \[ D = 0^2 - 4(-6)(-108) = 0 - 2592 = -2592 \]

Since the discriminant \(D\) is negative, this indicates there are no real solutions.

Therefore, the correct statement is: There will be no real solutions since the discriminant is negative.