For what positive integers $c$, with $c < 100$, does the following quadratic have rational roots? \[ 3x^2 + 20x + c \]

Guythatneedshelp Jun 1, 2023

#1**0 **

For a quadratic to have rational roots, its discriminant must be a perfect square. The discriminant of the given quadratic is 400 - 12c. For this to be a perfect square, we must have 400 - 12c = k^2 for some positive integer k. Solving, we get c = 400/12 - k^2. Since c < 100, we see that k must be 1, 2, 3, or 4. This gives us the four possible values of c: 25, 20, 15, and 12.

(Note: The quadratic may have other roots, such as irrational roots, that are not integers. We are only concerned with whether or not it has rational roots.)

Guest Jun 1, 2023