When Are Two Quadratic Functions Identical?

Two quadratic functions are identical simply when the ratio between the coefficients of the two functions are the same; one is a "multiple" of another. This is because, given a quadratic function $ax^2+bx+c=0$, we can multiply the entire equation by some constant $k$ to form the new quadratic function $akx^2+bkx+ck=0k=0$. The roots of the original function are, by the quadratic formula, $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ and the roots of the second function are $\frac{-bk\pm\sqrt{(bk)^2-4akck}}{2ak} = \frac{-bk\pm\sqrt{k^2(b^2-4ac)}}{2ak} = \frac{-bk\pm k\sqrt{b^2-4ac}}{2ak} = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$ which are simply the same roots as our original equation. We could also logically see that one equation could be divided by some constant $k$ to form the other, and the division of $k$ on both sides should not change the value of the roots.