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?
There are two non-real solutions.
There are two irrational solutions.
There are two rational solutions.
There are two identical solutions.
To use the quadratic formula, we have the quadratic polynomial in the form of ax^2 + bx + c = 0. Comparing the given quadratic polynomial -4x^2 - 24x - 36 = 0 to this form, we have a = -4, b = -24, and c = -36.
The quadratic formula is given by x = (-b ± √(b^2 - 4ac)) / (2a).
Plugging in the values, we have x = (-(-24) ± √((-24)^2 - 4(-4)(-36))) / (2(-4)).
Simplifying further, we have x = (24 ± √(576 - 576)) / (-8).
Since 576 - 576 = 0, the expression (√(576 - 576)) becomes √0 = 0.
So, the expression becomes x = (24 ± 0) / (-8).
This gives us x = 24 / -8 = -3 and x = -24 / -8 = 3.
Therefore, the solution set is {-3, 3}. Since both solutions are identical (3 = -(-3)), the correct statement is:
There are two identical solutions.