Question

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

Answer 1

To find the values of c for which the quadratic equation has rational roots, we need to consider the discriminant. The discriminant is the expression inside the square root in the quadratic formula. By checking the values of c from 1 to 99, we can determine which values make the expression a perfect square.

Step-by-step explanation:

To find the values of c for which the quadratic equation has rational roots, we need to consider the discriminant. The discriminant is the expression inside the square root in the quadratic formula. For the quadratic equation 3x^2 + 20x + c = 0, the discriminant is 20^2 - 4(3)(c).

For the roots to be rational, the discriminant must be a perfect square. So, 20^2 - 4(3)(c) must be a perfect square. We can simplify this as 400 - 12c. By checking the values of c from 1 to 99, we can determine which values make the expression a perfect square.

For example, when c = 32, the expression becomes 400 - 12(32) = 16, which is a perfect square. So, one possible value of c is 32. By checking all the values of c from 1 to 99, we can find the values that make the expression a perfect square.