Let a, b, c be real numbers, $$a \ne 0$$. If $$\alpha \,$$ is a root of $${a^2}{x^2} + bx + c = 0$$. $$\beta \,$$ is the root of $${a^2}{x^2} - bx - c = 0$$ and $$0 < \alpha \, < \,\beta $$, then the equation $${a^2}{x^2} + 2bx + 2c = 0$$ has a root $$\gamma $$ that always satisfies