How do you write a quadratic formula with $a, b$, and $c$ being integers and its solutions being rational numbers that are not integers?

Question

This question has me stumped and I feel like I must be missing something obvious. How do you write a quadratic equation in the standard form $ax^2+bx+c=0$ such that $a,b$ and $c$ are integers, but the solutions are rational numbers that are not integers?

Answer 1

Take two rational numbers, say $p$ and $q$. These are the roots of the equation $(x-p)(x-q)=0$. To get integer coefficients, just clear denominators, that is multiply both sides by the product of the denominators of $p$ and $q$. Then multiply out the left-hand side to get the equation in the desired form.