1. **Stating the problem:** We have a quadratic function $F(x) = ax^2 + bx + c$ with $a \neq 0$ and $x \in \mathbb{R}$. The graph of this function is a parabola intersecting the x-axis at two points $x_1$ and $x_2$. 2. **Formula used:** The quadratic can be factored as: $$F(x) = a(x - x_1)(x - x_2)$$ where $x_1$ and $x_2$ are the roots (zeros) of the quadratic. 3. **Important rules:** - Since $a \neq 0$, the parabola opens upwards if $a > 0$ and downwards if $a < 0$. - The roots $x_1$ and $x_2$ satisfy the quadratic equation $ax^2 + bx + c = 0$. 4. **Intermediate work:** - The roots can be found using the quadratic formula: $$x_1, x_2 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ - Once $x_1$ and $x_2$ are found, the quadratic can be expressed in factored form as above. 5. **Explanation:** The factored form $a(x - x_1)(x - x_2)$ shows the parabola's zeros explicitly. The sign of $a$ determines the direction the parabola opens. This form is useful for graphing and understanding the roots. **Final answer:** $$F(x) = a(x - x_1)(x - x_2)$$ with $x_1, x_2 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ and $a \neq 0$.