Two Solutions for All Quadratic Equations Quick Check

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

There will be two real solutions since the discriminant is positive.
There will be two real solutions since the discriminant is positive.
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Answer 1

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

The given equation can be rewritten as: \[ -6x^2 + 0x - 108 = 0 \] Here, \(a = -6\), \(b = 0\), and \(c = -108\).

Next, we calculate the discriminant using the formula: \[ D = b^2 - 4ac \]

Substituting the values of \(a\), \(b\), and \(c\): \[ D = 0^2 - 4(-6)(-108) \] \[ D = 0 - 4 \cdot 6 \cdot 108 \] \[ D = -2592 \]

The discriminant \(D\) is negative (\(-2592 < 0\)).

When the discriminant is negative, there are no real solutions to the equation.

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