There are infinitely many quadratic equations with roots (or zeros) of 1 and 5. They are k(x^2-6x+5)=0 for any non zero k. In order to make the solution unique, so you need to say "Find a quadratic equation whose coefficient of x^2 is 1 and roots are 1 and 5", for example.
This theorem is easy to understand but there is one more thing that when we multiply by a, this generates a family of quadratics with the same roots r1and r2 in general.
Do mean by the theorem, if the given roots are rational we need to multiply the factors with their LCDs to get into a standard form of quadratic equation ?