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 identical solutions There are two non-real solutions. There are two rational solutions. There are two irrational solutions.

Question

GPT-4o mini · Answer

To solve the quadratic equation $-4x^2 - 24x - 36 = 0$ using the quadratic formula, we first need to identify the coefficients in the standard form $ax^2 + bx + c = 0$. For the equation $-4x^2 - 24x - 36 = 0$, - $a = -4$ - $b = -24$ - $c = -36$ The quadratic formula is given by: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ Now, we can calculate the discriminant $(b^2 - 4ac)$: $$ b^2 = (-24)^2 = 576 $$ $$ 4ac = 4 \cdot (-4) \cdot (-36) = 576 $$ $$ b^2 - 4ac = 576 - 576 = 0 $$ Since the discriminant is equal to zero, we will have one repeated (or identical) solution. Now we can plug the values into the quadratic formula: $$ x = \frac{-(-24) \pm \sqrt{0}}{2(-4)} $$ $$ x = \frac{24 \pm 0}{-8} $$ $$ x = \frac{24}{-8} = -3 $$ Thus, the solution to the equation $-4x^2 - 24x - 36 = 0$ is: $$ x = -3 $$ Since we find one solution with multiplicity 2, we can conclude that there are two identical solutions. Finally, among the provided options, the accurate statement is: **There are two identical solutions.**