Question

Show that the Fundamental Theorem of Algebra is true for the quadratic polynomial −4x2−24x−36=0
by using the quadratic formula. Which of the following statements accurately describes the solution set?(1 point)
Responses

There are two irrational solutions.
There are two irrational solutions.

There are two non-real solutions.
There are two non-real solutions.

There are two identical solutions.
There are two identical solutions.

There are two rational solutions.

Answer 1

To show that the Fundamental Theorem of Algebra is true for the polynomial \( -4x^2 - 24x - 36 = 0 \), we will use the quadratic formula, which is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the quadratic polynomial \( -4x^2 - 24x - 36 \), the coefficients are:
- \( a = -4 \)
- \( b = -24 \)
- \( c = -36 \)

First, we need to calculate \( b^2 - 4ac \):

\[
b^2 = (-24)^2 = 576
\]

\[
4ac = 4 \cdot (-4) \cdot (-36) = 576
\]

Now we find the discriminant:

\[
b^2 - 4ac = 576 - 576 = 0
\]

Since the discriminant is zero, we can conclude that there is one repeated (identical) solution.

Now, we will compute the solution using the quadratic formula:

\[
x = \frac{-(-24) \pm \sqrt{0}}{2 \cdot -4} = \frac{24 \pm 0}{-8} = \frac{24}{-8} = -3
\]

Thus, the solution to the equation \( -4x^2 - 24x - 36 = 0 \) is \( x = -3 \), and this is a repeated solution.

### Conclusion

The original polynomial has one repeated solution:
- There are two identical solutions of \( x = -3 \).

Therefore, the correct statement describing the solution set is:

**There are two identical solutions.**