To show that the Fundamental Theorem of Algebra is true for the quadratic polynomial \( -4x^2 - 24x - 36 = 0 \), we will use the quadratic formula, which states:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, our coefficients from the polynomial \( -4x^2 - 24x - 36 \) are:
- \( a = -4 \)
- \( b = -24 \)
- \( c = -36 \)
Now, we can plug in these values into the quadratic formula.
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Calculate the discriminant \( b^2 - 4ac \): \[ b^2 = (-24)^2 = 576 \] \[ 4ac = 4(-4)(-36) = 576 \] Therefore, the discriminant is: \[ b^2 - 4ac = 576 - 576 = 0 \]
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Substitute \( a, b, \) and the discriminant into the quadratic formula: \[ x = \frac{-(-24) \pm \sqrt{0}}{2(-4)} \] This simplifies to: \[ x = \frac{24 \pm 0}{-8} = \frac{24}{-8} = -3 \]
The solutions to the polynomial equation \( -4x^2 - 24x - 36 = 0 \) is: \[ x = -3 \]
Since the discriminant is zero, this indicates that there are two identical solutions. Therefore, the solution set accurately described is:
There are two identical solutions.