Basics and Methods of Solving Quadratic Equations

Basics and Methods of Solving Quadratic Equations

This lesson covers the following key concepts:

• Standard form of quadratic equation: ax² + bx + c = 0
• Methods: factorization, completing the square, quadratic formula
• Discriminant and nature of roots
• Relation between roots and coefficients
• Formation of quadratic equations with given roots
• Common mistakes and edge cases

Important Formulas

• $$ax^2 + bx + c = 0, a \neq 0$$
• $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
• $$D = b^2 - 4ac \text{ (discriminant)}$$
• $$\alpha + \beta = -\frac{b}{a}, \alpha\beta = \frac{c}{a}$$
• $$x^2 - (\alpha + \beta)x + \alpha\beta = 0$$
• $$\text{Completing square: } a\left(x + \frac{b}{2a}\right)^2 + \frac{4ac - b^2}{4a} = 0$$