Quadratic Equation 47F2Ed

1. **Stating the problem:** We are given a quadratic equation $$a x^2 + b x + c = 0$$ where $a$, $b$, and $c$ are constants and $a \neq 0$. 2. **Formula used:** The solutions to the quadratic equation are found using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a}$$ 3. **Important rules:** - The discriminant $$\Delta = b^2 - 4 a c$$ determines the nature of the roots. - If $$\Delta > 0$$, there are two distinct real roots. - If $$\Delta = 0$$, there is one real root (a repeated root). - If $$\Delta < 0$$, there are two complex conjugate roots. 4. **Intermediate work:** - Calculate the discriminant $$\Delta = b^2 - 4 a c$$. - Substitute into the quadratic formula: $$x = \frac{-b \pm \sqrt{\Delta}}{2 a}$$ 5. **Explanation:** The quadratic formula gives the values of $x$ that satisfy the equation. The term under the square root, called the discriminant, tells us how many and what type of solutions exist. Dividing by $2a$ normalizes the equation since the highest degree term is $a x^2$. This formula works for any quadratic equation where $a \neq 0$.