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?

There are two non-real solutions.

There are two irrational solutions.

There are two rational solutions.

There are two identical solutions.

Answer 1

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.