If $\alpha$ and $\beta$ are the roots of the quadratic equation $a{x}^2+2bx+c=0$ and $\alpha +\gamma$, $\beta +\gamma$ are the roots of $A{x}^2+2Bx+C=0$, then which one of the following is correct?

The difference between the roots of the first equation is:
$$(\alpha )-(\beta )=\alpha -\beta$$

The difference between the roots of the second equation is:

$$(\alpha +\gamma )-(\beta +\gamma )=\alpha -\beta$$

Since the difference between the roots is identical for both quadratic equations, their absolute differences squared must also be equal:

$$(\alpha -\beta {)}^2=[(\alpha +\gamma )-(\beta +\gamma ){]}^2$$

For any quadratic equation $p{x}^2+qx+r=0$, the square of the difference of roots is given by $\frac{q^2-4pr}{p^2}$.

Applying this property to both equations:

1. For $a{x}^2+2bx+c=0$:

$$(\alpha -\beta {)}^2=\frac{(2b{)}^2-4ac}{a^2}=\frac{4b^2-4ac}{a^2}=\frac{4(b^2-ac)}{a^2}$$

2. For $A{x}^2+2Bx+C=0$:

$$[(\alpha +\gamma )-(\beta +\gamma ){]}^2=\frac{(2B{)}^2-4AC}{A^2}=\frac{4B^2-4AC}{A^2}=\frac{4(B^2-AC)}{A^2}$$

Equating the two expressions:

$$\frac{4(b^2-ac)}{a^2}=\frac{4(B^2-AC)}{A^2}$$

$$\frac{b^2-ac}{a^2}=\frac{B^2-AC}{A^2}$$

$$A^2(b^2-ac)=a^2(B^2-AC)$$