To verify the Fundamental Theorem of Algebra for the quadratic polynomial \(-4x^2 - 24x - 36 = 0\) using the quadratic formula, we will first identify the coefficients \(a\), \(b\), and \(c\):
- \(a = -4\)
- \(b = -24\)
- \(c = -36\)
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
We will start by calculating the discriminant \(\Delta = b^2 - 4ac\):
\[ b^2 = (-24)^2 = 576 \] \[ 4ac = 4(-4)(-36) = 576 \] \[ \Delta = 576 - 576 = 0 \]
Next, we substitute the values into the quadratic formula:
\[ x = \frac{-(-24) \pm \sqrt{0}}{2(-4)} = \frac{24 \pm 0}{-8} \]
This simplifies to:
\[ x = \frac{24}{-8} = -3 \]
Since the discriminant is \(0\), there is exactly one unique solution, which confirms that the two solutions are identical.
Thus, the solution set for the quadratic polynomial \(-4x^2 - 24x - 36 = 0\) is:
\[ {-3} \]
Now, considering the options in relation to the solution set, the correct statement is:
There are two identical solutions.