Quadratic Equations

Overview

Quadratic equations form a cornerstone topic in the CG TET Paper II Mathematics section. These equations appear in various forms—direct solving, word problems, and questions testing the relationship between roots and coefficients. Mastering this topic builds a foundation for higher algebra and helps in solving real-world problems involving areas, projectile motion, and optimization.

For CG TET, you must be able to identify quadratic equations, solve them using multiple methods (factorisation, formula, completing the square), determine the nature of roots using the discriminant, and apply the sum-product relationships of roots. Questions typically test computational accuracy and conceptual clarity, so both speed and understanding matter equally.

Key Concepts

• **Standard Form**: A quadratic equation is written as ax² + bx + c = 0, where a ≠ 0 and a, b, c are real numbers. The condition a ≠ 0 is essential—otherwise it becomes linear.

• **Roots/Solutions**: Values of x that satisfy the equation. A quadratic equation has exactly two roots (which may be equal, distinct, or complex).

• **Discriminant (D)**: The expression D = b² − 4ac determines the nature of roots without actually solving the equation.

• **Methods of Solving**: Four standard approaches exist—factorisation, quadratic formula, completing the square, and graphical method. For exams, factorisation and formula are most frequently tested.

• **Sum and Product of Roots**: If α and β are roots, then α + β = −b/a and αβ = c/a. These relationships allow you to form equations or find expressions involving roots.

• **Symmetric Functions**: Expressions like α² + β², α³ + β³, and 1/α + 1/β can be computed using sum and product without finding individual roots.

Formulas / Key Facts

| Formula | Context | |---------|---------| | ax² + bx + c = 0 | Standard form of quadratic equation | | x = (−b ± √(b² − 4ac)) / 2a | Quadratic formula (Shreedharacharya's rule) | | D = b² − 4ac | Discriminant formula | | D > 0 → Two distinct real roots | Nature of roots | | D = 0 → Two equal real roots | Both roots are −b/2a | | D < 0 → No real roots | Roots are complex conjugates | | α + β = −b/a | Sum of roots | | αβ = c/a | Product of roots | | x² − (α + β)x + αβ = 0 | Equation with roots α and β | | α² + β² = (α + β)² − 2αβ | Derived symmetric function | | α − β = ±√D / a | Difference of roots |

Worked Examples

**Example 1: Solve by Factorisation**

Solve: x² − 5x + 6 = 0

Step 1: Find two numbers whose product is 6 and sum is −5. Numbers are −2 and −3 (since −2 × −3 = 6 and −2 + −3 = −5)