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 one real solution 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.

1 answer

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

In this case:

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

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

Now, we substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula:

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

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

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