Question

two solutions for all quadratic equations quick check
use the discriminant to determine the number of real solutions of $-6x^2 - 108 = 0$. which of the following statements gives the correct explanation? (1 point)

• 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 one real solution since the discriminant is zero.
• there will be no real solutions since the discriminant is negative.

Explanation:

Step1: Recall discriminant formula

For quadratic equation \(ax^2 + bx + c = 0\), discriminant \(D = b^2 - 4ac\).
Given equation: \(-6x^2 - 108 = 0\), so \(a = -6\), \(b = 0\), \(c = -108\).

Step2: Calculate discriminant

Substitute values into \(D\):
\(D = 0^2 - 4\times(-6)\times(-108)\)
\(D = 0 - 4\times6\times108\)
\(D = -2592\)

Step3: Analyze discriminant

Since \(D = -2592 < 0\), a negative discriminant means no real solutions (real solutions exist only when \(D \geq 0\)).

Answer:

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