Quadratic Equation B4D6Af

1. **Problem Statement:** A quadratic equation is a polynomial equation of degree 2, generally written as $$ax^2 + bx + c = 0$$ where $a$, $b$, and $c$ are constants and $a \neq 0$. 2. **Formula and Important Rules:** The solutions to the quadratic equation can be found using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, the term under the square root, $$\Delta = b^2 - 4ac$$, is called the discriminant. 3. **Types of Quadratic Equations Based on Discriminant:** - If $$\Delta > 0$$, there are two distinct real roots. - If $$\Delta = 0$$, there is exactly one real root (a repeated root). - If $$\Delta < 0$$, there are two complex conjugate roots. 4. **Step-by-Step Solution Example:** Solve $$2x^2 - 4x - 6 = 0$$. Step 1: Identify coefficients: $$a=2$$, $$b=-4$$, $$c=-6$$. Step 2: Calculate discriminant: $$\Delta = (-4)^2 - 4 \times 2 \times (-6) = 16 + 48 = 64$$. Step 3: Since $$\Delta > 0$$, two distinct real roots exist. Step 4: Apply quadratic formula: $$x = \frac{-(-4) \pm \sqrt{64}}{2 \times 2} = \frac{4 \pm 8}{4}$$. Step 5: Calculate roots: $$x_1 = \frac{4 + 8}{4} = \frac{12}{4} = 3$$ $$x_2 = \frac{4 - 8}{4} = \frac{-4}{4} = -1$$ 5. **Summary:** Quadratic equations can be solved by factoring (if possible), completing the square, or using the quadratic formula. The discriminant helps determine the nature of the roots. This explanation covers the basics and types of quadratic equations and how to solve them step-by-step.