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

Answer 1

To solve the quadratic polynomial \(-4x^2 - 24x - 36 = 0\) using the quadratic formula, we start by identifying the coefficients. The quadratic formula is given by:

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

For the polynomial \(-4x^2 - 24x - 36\), we can identify:
- \(a = -4\)
- \(b = -24\)
- \(c = -36\)

Now, we will calculate the discriminant \(D\) first:

\[
D = b^2 - 4ac
\]

Substituting in the values of \(a\), \(b\), and \(c\):

\[
D = (-24)^2 - 4(-4)(-36)
\]
\[
D = 576 - 576
\]
\[
D = 0
\]

Since the discriminant \(D\) is \(0\), this indicates that there is one unique solution (a repeated root) for the quadratic equation.

Next, we can apply the quadratic formula:

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

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

Finally, since the quadratic equation has a discriminant of zero, it has two identical solutions. Hence, the accurate description of the solution set is:

**There are two identical solutions.**