Question

For what positive integers $c$, with $c < 100$, does the following quadratic have rational roots?

\[ 3x^2 + 20x + c \]
If you find more than one, then list the values separated by commas.

Answer 1

Answer: A quadratic of the form $ax^2 + bx + c$ has rational roots if and only if its discriminant, $b^2 - 4ac$, is a perfect square. In this case, the discriminant is $(20)^2 - 4(3)(c) = 400 - 12c$. So, we need to find the positive integers $c < 100$ that make $12c$ a perfect square.

The perfect squares that are less than $100$ are $1,4,9,16,25,36,49,64,81$, so the possible values for $c$ are $33,44,55,66,77,88$.

So, the positive integers $c$, with $c < 100$, that make the quadratic have rational roots are $33,44,55,66,77,88$

Explanation: