Question

The princess examines one plant and can’t seem to locate its roots. The plant is labeled as p(x)=x^2+x+3 Which values should the princess test to see if the plant has rational roots? There may be more than one correct answer. Select all correct answers
1/3
-1/3
-1
1
-3
3
0
The princess finished classifying almost all of the plants and matching them to roots with the exception of the plant mentioned in the previous problem and one other plant. There is only one root left unmatched, but she can’t identify it.
Explain whether or not the plant labeled p(x)=x^2+x+3 has any rational or irrational roots

Answer 1

Let's actually find the roots, using the quadratic formula:

p(x)=x^2+x+3 gives us a=1, b=1 and c=3.

                 -1 plus or minus sqrt(1^2-4(1)(3))
Then x = -----------------------------------------------          
                                           2

The discriminant here is negative, so the roots x will be complex:

              -1 plus or minus sqrt(-11)      -1 plus or minus i*sqrt(11)
       x = ---------------------------------- = -------------------------------------
                          2                                                 2

These are irrational roots; they cannot be expressed as the ratios of integers.