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)ResponsesThere 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.There will be no real solutions since the discriminant is negative.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 two real solutions since the discriminant is positive.

Question

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)ResponsesThere 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.There will be no real solutions since the discriminant is negative.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 two real solutions since the discriminant is positive.

GPT-4o mini · Answer

To determine the number of real solutions for the equation \(-6x^2 - 108 = 0\), we first rearrange it in the standard quadratic form \(ax^2 + bx + c = 0\).

The equation can be rewritten as: \[ -6x^2 = 108 \implies 6x^2 = -108 \] However, reversing the sign gives: \[ 6x^2 + 108 = 0 \]

Identifying coefficients:

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

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

Now, substituting in the values we have: \[ D = 0^2 - 4(6)(108) = 0 - 2592 = -2592 \]

Since the discriminant \(D\) is negative, we conclude that there are no real solutions for the equation.

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