show the 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?
there are two irrational solutions
there are two identical solutions
there are two rational solutions
there are two non-real solutions
In order to apply the quadratic formula, let's consider the quadratic equation -4x^2 - 24x - 36 = 0.
The quadratic formula states that for a quadratic equation in the form ax^2 + bx + c = 0, the solutions can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our quadratic equation -4x^2 - 24x - 36 = 0, we can determine that a = -4, b = -24, and c = -36.
Using the quadratic formula:
x = (-(-24) ± √((-24)^2 - 4(-4)(-36))) / (2(-4))
x = (24 ± √(576 - 576)) / (-8)
x = (24 ± √0) / (-8)
x = (24 ± 0) / (-8)
x = 24 / (-8)
x = -3
So, the equation -4x^2 - 24x - 36 = 0 has two identical solutions: x = -3.
Thus, the correct statement is: "There are two identical solutions."