Use the discriminant to determine the number of real solutions of −6x2−108=0

Question

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

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 of the equation \(-6x^2 - 108 = 0\) using the discriminant, we can first rewrite the equation in standard quadratic form, which is \(ax^2 + bx + c = 0\).

Here, we have:

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

The discriminant \(D\) is calculated using the formula: \[ D = b^2 - 4ac \]

Plugging in the values we have: \[ D = 0^2 - 4(-6)(-108) \] \[ D = 0 - 4 \cdot 6 \cdot 108 \] \[ D = -864 \]

Since the discriminant \(D\) is negative (\(D < 0\)), it means that the quadratic equation has no real solutions.

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