Question

show that the fundamental theorem of algebra is true for the quadratic polynomial $-4x^2 - 24x - 36 = 0$ by using the quadratic formula. which of the following statements accurately describes the solution set? (1 point)

there are two rational solutions.

there are two non - real solutions.

there are two irrational solutions.

there are two identical solutions.

Explanation:

Step1: Recall Quadratic Formula

The quadratic formula for a quadratic equation \(ax^2 + bx + c = 0\) is \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\). For the equation \(-4x^2-24x - 36 = 0\), we have \(a=-4\), \(b = - 24\), and \(c=-36\).

Step2: Calculate the Discriminant

The discriminant is \(D=b^2 - 4ac\). Substitute the values: \(D=(-24)^2-4\times(-4)\times(-36)\). Calculate \((-24)^2 = 576\), \(4\times(-4)\times(-36)=4\times144 = 576\). So \(D = 576-576=0\).

Step3: Analyze the Discriminant

When the discriminant \(D = 0\), the quadratic formula gives \(x=\frac{-b\pm\sqrt{0}}{2a}=\frac{-b}{2a}\) (since \(\sqrt{0}=0\)). So we have a repeated root (two identical solutions) because the \(\pm\) part doesn't change the value (both solutions are \(\frac{-(-24)}{2\times(-4)}=\frac{24}{-8}=- 3\)).

Answer:

There are two identical solutions.