Show that the Fundamental Theorem of Algebra is true for the quadratic polynomial x2−16x+39=0 through solving by factoring. Which of the following statements accurately describes the solution set?(1 point)
Responses
There are two non-real solutions.
There are two non-real solutions.
There are two identical solutions.
There are two identical solutions.
There are two irrational solutions.
There are two irrational solutions.
There are two rational solutions.
To solve the quadratic equation x^2 - 16x + 39 = 0 by factoring, we need to find two numbers whose product is 39 and whose sum is -16.
By analyzing the factors of 39, we see that the possible factor pairs are (1, 39), (-1, -39), (3, 13), and (-3, -13).
Out of these options, the factor pair that gives us a sum of -16 is (-3, -13).
Therefore, we can rewrite the equation as follows:
(x - 3)(x - 13) = 0
To find the solutions, we set each factor equal to zero and solve:
x - 3 = 0 ---> x = 3
x - 13 = 0 ---> x = 13
Hence, the solution set for the equation x^2 - 16x + 39 = 0 is {3, 13}.
Based on the solution set, the correct statement is:
There are two rational solutions.