The quadratic equation with integer coefficients whose roots are 1/2 and −2/3 is:

The sum of the roots is 1/2 − 2/3 = −1/6 and the product is −1/3. So $x^2+x/6-1/3=0$, and multiplying through by 6 gives $6x^2+x-2=0$, which factorises as $(2x-1)(3x+2).(6x^2-x-2=(3x-2)(2x+1)$ has the roots 2/3 and −1/2 — the signs attached to the wrong root.)