To determine the nature of the solutions for the quadratic equation \(-4x^2 - 24x - 40 = 0\), we can analyze the equation using the discriminant from the quadratic formula.
The standard form of a quadratic equation is \(ax^2 + bx + c = 0\), where:
- \(a = -4\)
- \(b = -24\)
- \(c = -40\)
The discriminant \(D\) is given by the formula: \[ D = b^2 - 4ac \]
Now, let's calculate it:
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Calculate \(b^2\): \[ b^2 = (-24)^2 = 576 \]
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Calculate \(4ac\): \[ 4ac = 4 \cdot (-4) \cdot (-40) = 16 \cdot 40 = 640 \]
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Now calculate the discriminant: \[ D = 576 - 640 = -64 \]
Since the discriminant \(D\) is negative (\(-64 < 0\)), this indicates that the quadratic equation has two imaginary solutions.
Therefore, the accurate description of the solution set for the equation \(-4x^2 - 24x - 40 = 0\) is:
C: There are two imaginary solutions.